Calibration of the STS-1 seismometer; preliminary results

PHYSICS

~

OFTHE EARTH AND P LAN ETA RY INTERIORS

_________

ELSEVIER

Physics of the Earth and Planetary Interiors 84 (1994) 299—304

Calibration of the STS-1 seismometer; preliminary results J.-F. Fels Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0225, USA (Received 1 January 1993; revision accepted 8 December 1993)

Abstract The cross-spectrum method has been applied to the random binary calibration of the STS-1 seismometers (VBB option) installed at the Incorporated Research Institutions for Seismology/Internal Deployment of Accelerometers (IRIS/IDA) stations. It proves to be highly efficient at reducing the effect upon the derived response of the ground noise present in the output signal. The residual noise is smaller than 0.5% in amplitude and 0.50 in phase, for frequencies ranging from 0.5 to 100 mHz, provided that data series as long as 12 h are available. Except for a few cases, we observe that the actual transfer function remains within 2% in amplitude and 1.2°in phase relative to the generally accepted model, also referred to as the ‘nominal’ transfer function. The deviations are attributed essentially to inaccurate values of the low-frequency corner of the response. We also have evidence of other small deviations owing to the setting of the mechanical free period of the seismometer upon installation in the field, and to the approximations introduced in the nominal model. We propose a corrected model which better describes the behavior of the instrument in the low-frequency range. The results presented here involve a relative calibration of the sensors; the determination of the absolute value of the sensitivity factor must be obtained from a different procedure.

1. Cross-spectrum analysis of the calibration signal The calibration input signal is a sequence of random binary voltage steps (Berger et a!., 1979), lasting for about 12 h, with time durations between steps which are multiples of 10 s (the ‘clock period’). The input calibration signal a~(t), equivalent to a ground acceleration, and the output signal y(t) are shown in Fig. 1. The instrument transfer function can be obtamed by the ratio between the Fourier transforms of the output and the input. However, a more convenient tool is to compare observed and expected transfer functions, which requires con-

volving the input signal with the expected transfer function, and calculate the relative error by dividing the observed frequency response Y by the expected X. Let A~be the Fourier transforms of the random input signal a~(t), N the transform of the total noise referred to the input, Ha the actual transfer function of the system and He the expected transfer function. Then

X

Ha(Ac +N) HeA c

— —

Ha

N

1 +

(1)

The effect of noise present in the output signal is greatly reduced by multiplying both spectra Y and X of Eq. (1) by the conjugate spectrum X’~

0031-9201/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0031-9201(94)05040-5

300

J.-F. Fels /Physics of the Earth and Planetary Interiors 84 (1994) 299—304

and then averaging these products. This is the principle of the cross-spectrum technique, in which we calculate the ratio between the averaged cross-spectra and the averaged synthetic output power:

102 to’ 10° to-1 10-2

2

1H

81

to-3

~

~

A5~N~

H~,(

I ~-4 I O-°

A~N

io-7

FIAcI2

(2)

108 10-°

The ratio R represents an estimate of the ratio actual/predicted transfer functions, and as the noise and the calibration signal are statistically independent, the residual noise 1V. is supposedly much smaller than the ratio N/AC. Fig. 2 shows plots of the various spectra (with arbitrary vertical offsets): the calibration input signal A~, the synthetic output power P~ after convolution, and the modulus of the cross-spectrum C,~,.As expected, the cross-spectrum and the synthetic output power are very similar and close to the modulus square of the expected transfer function He in the low-frequency range. The data flow chart of Fig. 3 illustrates the successive steps of the cross-spectrum method.

y(n)

to—2 to—’ 100 Hertz Fig. 2. Spectra involved in the cross-spectrum method and modulus square of the expected transfer function.

2. Results using the nominal transfer function The ratio R has been computed for 12 STS-1 sensors, according to Eq. (2), where the nominal transfer function was used to convolve the input calibration signal. The input—output time series were segmented into 15 consecutive sections of 2000 s, with a cosine taper and an overlap of 50%. The relative errors are plotted as R 1 in Fig. 4. Deviations relative to the predicted transfer function occur principally around the lowfrequency corner (2.77 mHz) and around the frequency of the suspension electronic compensator (25 mHz). The latter are caused by the pendulum free period set to values smaller than approximately 40 s. This feature, unaccounted for in the nominal model of the transfer function, is briefly analyzed below. —

observed

3. STS-1 Parametric sensor and approximate models of the

o

100

I

200 300 400 500 Seconds Fig. 1. Random binary calibration of STS-1 seismometers. Time series waveforms, passed through a decimation filter,

We will define three models: the parametric model T~deduced from the block diagram shown in Fig. 5; the approximate ‘nominal’ model TA as described in the manual; a corrected model T~. The analytic expressions of the three models are as follows:

J.-F. Eels /Physics of the Earth and Planetary Interiors 84 (1994) 299—304

parametric model:

corrected model:

+f

=

2(s +f 2(s ~6 + ~5~5 +s~4~4 +4) ~ 3 + ~2~2 +

3)

3s

2

~

2+

1s + c0

T~

=

2

s

2

+

2b1 fis + f1

2

s +

2b

2f2s

Actual transfer function

2

+f~s

+f 2 4) 5f5s +f~

(s + 2b

s 2 + 2b X s

2f2s +f~

The models T~and T~are actually equivalent, the only difference being the factorization of the

f22 ~

Sensor output

s

2bs 1f1s

+f3

approximate model (‘nominal’, from the manufacturer’s manual): TA =

301

Calibration signal

(~)

Time align

Sensors models

______

IiteH

[~~e

I

r~i

Convolve

Average Cross-spectrum -~ Y

(~-—-—-~) transfer function

j~

Power

p~

H~

Plot

R- 1

Adjust parameters

~=Th~ Fig. 3. Data flow chart of the cross-spectral analysis and estimation of the actual transfer function.

302

i-F. Fels /Physics of the Earth and Planetary Interiors 84 (1994) 299—304

sixth-order polynomial of T~into the three second-order polynomials, leading to the poles and zeroes form T~. The corrected model T~describes more accurately the actual behavior of the instrument, as predicted by a parametric analysis, in particular the pendulum. influence of mechanical parameters of the It the is characterized by the following features: the double pole f4 corresponding to the suspension electronic compensator located at the level of the displacement transducer (Wielandt et a!., 1986), and the double pole f5, b~corresponding to the mechanical sus-

A1~ 1992—100 Z

pension, are no longer eliminated from the equation. Also, the double poles f1, b~and f5, b5 depend upon the suspension parameters fm’ bm (note that the product bm fm is a mechanical characteristic which is constant for a given pendu2/(s2 lum). + 2b In the model TA, the fraction (s +f4) 5f5s + f~)has been approximated by one, therefore ignoring the influence of the mechanical suspension. Fig. 6 illustrates the differences between the nominal model TA and the parametnc model T~,and also the effect on T~of1/fm~ various It values of the mechanical period Tm

z

—~~~____._____..~,

N_ E

E

—

PFO 1992—068 Z

_________________________________________

Z

~

N_~

E

E

R~N 1992—344 Z

Z

:~ TLY 1992—187 Z N

:~

—_~—-—-“

Z

—__~__------—-—-——-~‘‘

N

~

E~-~

io—~

_—-----------—--~—-------~

io—’

10_2

Hertz Misfit on modulus

10_i

Hertz 2 percent

Misfit on phese

1.2 degree

Fig. 4. Errors on observed transfer functions relative to the nominal model of the transfer function.

i-F. Eels/Physics of theEarth and Planetary Interiors 84 (1994) 299—304

2~e~tion

A 0

1 1

Displacementtransducer (including produlum compensator)

Pendulum

~\T”I

F—tIII~II1—--

T1

303

=

1+ ~, ~‘2(T3T4+ T5T6)

Proportionaland derivativzfezdback

:~:‘

Forcer 1 (0k)

~ L[~

T

—

1

—

Forcer 2 (°2)

(3

1

s+d VL,,

7201 1

T5=

Fig. 5. Block diagram of the STS-1 system and transfer functions of the sub-systems.

should be noted that use of negative values of the mechanical period Tm is a short way to indicate that the pendulum is mechanically unstable.

4. Estimation of the actual transfer function using the corrected model The examination of the model T~shows that we have four independent variables (as opposed to two in the nominal model), which can be used to minimize the error deduced from the crossspectral analysis. These variables are: ~ b1, describing the low-frequency corner of the Fesponse; f5, b5, describing the combined effect of the suspension compensator and the suspension parameters. As the expressions of f5, b5 cannot 1.03

~

“~

5. Conclusions The proposed corrected model T~, whose number of poles and zeroes equal those of the parametric model, allows us to explain almost all of the small deviations in the actual transfer

~‘‘—“•~

t 02

.

.

—

1 Ot

Q97I1

be established analytically with respect to the physical parameters, we use instead fm, bm, which are entered into the calculation of the coefficients of T~.The roots of the denominator polynomial are computed by means of a standard algorithm, and then converted into the poles and zeroes of the form T~.The residual errors after adjustment of the above variables are plotted in Fig. 7.

bJ,n

I

001

(fIr)

1’,,,

—20

II~I

“

_______

—

—

T

(sgc~,

I

_______

T

_

io’ Hertz Fig. 6. Comparison between the modulus of the nominal model TA with the parametric model T~,for a damping constant equal to 0.01 and several values of the mechanical period Tm. Curves are plotted with arbitrary vertical offsets.

bmfm

304

i-F. Eels /Physics of the Earth and Planetary Interiors 84 (1994) 299—304

ALE 1992—100 Z

E

:~

~

___________—.

________

PFO 1992—069 2

..-

N

TLY 1992—187 Z

—..~tc7.’.--—

..~._

2

_________________________________________

N

E

N

—~—-_~a.Ia~,.c._-.

E

RPN 1992—344 2

Z

~_

~__~_.—_-_~~ ..__—__

N

—

__~—~

E

—.——

____________

,.__....

Z

_______________________________________

~_...,-

N

____—__-________.~_-_.—~

S

________

10—a

.

10’

io—~

Hertz Misfit en modulus

___________________

____

S

~_.

N

io—~

2

—~

~~_~_—

10—2

lI_I

Hertz 2 percent

Misfit on phase

1.2 degree

Fig. 7. Misfits of the observed transfer functions with the estimated transfer functions, calculated with the corrected model T~.

function of STS-1 seismometers without complicating significantly its analytic expression. In the best cases, the misfit of the observed response with the estimated response is of the order of 0.1% in amplitude and 0.1% in phase. If the mechanical period of the suspension is not set to a value larger than approximately 40 s, the instrument transfer function may be affected significantly in the frequency range 10—100 mHz. A few calibrations performed 2 years apart on the same sensors seem to indicate that the stability of the response over time is better than 0.5%. In other words the effects of aging on the electronic and mechanical parameters are small. The calibration of the instruments in the high-frequency range (above 0.2 Hz), which requires a clock period of

the random binary signal equal to or higher than 0.05 s, will be implemented in the future.

References Berger, J., Agnew, D., Parker, R. and Farell, W., 1979. Seismic system calibration: 2. Cross-spectral calibration using random binary signal. Bull. Seismol. Soc. Am., 69: 271—288. Fels, J.-F. and J. Berger. Parametric analysis and calibration of the STS-1 seismometer ofthe IRIS/IDA network. Bull. Seismol. Soc. Am., submitted. Wielandt, E. and Steim, J., 1986. A digital very broadband seismograph. Ann. Geophys., 4B: 227.