Applied Geophysics : Seismics

Response Functions of Seismographs

Applet ( in separate window, ca. 480 x 530 Pixel )

The applet shows
  amplitude and phase of the transferfunction with respect to ground displacement, velocity and acceleraton,
and
  the response in the time domain to some impulsive ground motions and to a sweep signal
for some historical instruments and actually used seismograph systems.

( Background : see Seismometer Documentaion,
Seismometer-Galvanometer Dokumentation and
Manual of Seismological Observatory Practice,
in separate window )

NEW : Determination of Systemparameters
  for an evtl. simplified transferfunction, applying the calculus of the program CALEX to the simulated recording of a broadband acceleration signal ( sweep ).

  see :
  Seismic Sensors and their Calibration by E. Wielandt und
  free software by E. Wielandt, ( description of program and FORTRAN Source calex.f )

- Table of Content -

> HowTo
>   Program Control
>   Graphics Control
> Screenshots
>   Frequency Domain
>   Time Domain
> CALEX-Calibration
>   Comments
>   Program Control
>   Results
> Download

- HowTo -

Dialogue Window :
( Screenshot )

SEISMOGRAPH SYSTEM : WIECHERT_Z...
alows to select one of the seismograph systems implemented :

> WIECHERT_Z / H and WOOD-ANDERSON
>   are historical seismometers with "direct" recording,
>   characterized by natural period, damping and a static amplification ( = recorded amplitude / mass excursion ),
>   where the feedback force of the mechanical recording system ( WIECHERT ) is neglected.
>   ( Parameter values from : Elemente der Seismologie und Seismometrie (1960), E.F.Sawarenski and D.P.Kirnos. )

> WWSSN_SP / LP, BENIOFF_LP, GALITZIN and KIRNOS
>   are seismometer - galvanometer systems, where
>   a velocity transducer ( coil-magnet-assemly ) is connected to a galvanometer coil via a resistive T element,
>   i.e. two resistances in series with seismometer / galvanometer coil and one shunt resistance.

>   The response of the system is charakterized by the natural period and the damping of both seismometer and galvanometer and
>   a dimensionless "coupling coefficient" σ^2 ( denoted α^2 in some publications, range 0...+1 ), describing the reciprocal influence of the two electro-mechanical devices.
>   ( see Seismometer-Galvanometer Documentation
>   and applet Seismometer-Galvanometer Coupling )

>   The frequency response corresponds to the product of a 2-pole high pass ( seismometer ) and a 2-pole low pass ( galvanometer )
>   with an adittional quadratic term proportional to σ^2 in the resulting denominator polynomial of 4th degree and a factor proportional to σ ( = sqrt(σ^2) ), determining the amplification of the system.

>   The parameter values for the systems implemented are based on tables in Manual of Seismological Observatory Practice ( 1979 Edition, Chap. Instruments, 1 Type of Instrument ).

>   These tables provide numerical values for natural period and damping of seismometers and galvanometers and for the "coupling coefficient" for various types of seismograph systems.
>   ( without details of the the instruments used and the coupling network )

>   For the systems simulated in the applet an amplification factor of the frequency response is chosen to meat the amplification curves shown in fig. 1.1 of the manual.

> For the seismometer S-13_1Hz ( Geotech )
>   the output voltage of the moving coil transducer is is observed on a recorder with unity gain in the relevant frequency range of 0 to 1000 [Hz].

> The geophones SM4_10Hz and L4_2Hz
>   are moving coil instruments connected to a digital recording system StrataView, and
>   the parameters used are taken from the data sheets of the manufacturers,
>   i.e. I/O Sensor Nederland bv ( SM4 ), Mark Products ( L4 ) and Geometrics ( StrataView ).

> The seismometer STS_2 ( Wielandt - Streckeisen )
>   is a force-balance broadband seismometer, and
>   its parameters are taken from the manual of the instrument and from Seismic Sensors and their Calibration, E. Wieland,
>   in New Manual of Observatory Practice.

RESP TO ...
selects the type of ground motion, i.e.
  displacement,
  velocity
  or
  acceleration
for the calculation of the transfer function ( frequency domain ) and the response in the time domain.

SIG^2 = 0.980 ... 1.E-8
  refers to seismometer - galvanometer systems only, and
  allows to vary the value of the coefficient σ^2, preserving the other parameters of the system, to demonstate the influence of the coupling coefficient on the response curve.

  The default value of σ^2 ( from definition table ) is displayed on white background, user selected values on yellow background.
  ( Details of seismometer-galvanometer coupling see : Seismometer-Galvanometer Documentation )
CLEAR TEXT ...
  clears the text area of the applet
  and lists
  the parameters of the actually selected system ( LIST SEIS ),
  the coeficients of the frequency response and / or of the corresponding Z-transform ( LIST COEFF ),
  or
  the frequency / time response ( FREQ / TIME RESP ).
  ( TIME RESP lists max. 500 samples starting from t = 0 [s], CONT TIME lists subsequent blocks of 500 SMP each. )

The graphic display is controlled by the display dialogue ( background light blue ) :

> XFR (F) / XFR (Z) / XF(BIL) / F + Z
>   shows the frequency response, the corresponding Z-transform or both
>   ( AMPL = ampltude, PHAS = phase in separate windows )
> for
>   SYS_OUT = system output,
>   SEIS = seismometer output and
>   GALV = transfer function of the galvanometer
>   ( HP_1 = 1-pole high pass for StrataView with SM4_10Hz and L4_2Hz ).

> HIDE RESP WIN ...
> hides / shows the above output signals in the time domain for different ground motions :

>   DELTA IMP ( = Dirac impulse ),
>   UNIT STEP,
>   KUEPPERS_2 ( = Kueppers signal, order 2 ),
>   K_2 x HANNING ( = as above, with Hanning taper ),
>   or a sweep signal ( DOWN / UP SWEEP ) covering the the range of characteristc frequencies a system.

> ( 200 SMP... refers to Kueppers signal only and allows to select the signal length and, together with F-SAMP, the predominant frequency of the signal. )

> F-SAMP : 1 ... 800 [Hz] ( = sampling frequency )
>   refers to the Z-transform XFR ( Z ), used as a recursive filter to calculate the response in the time domain,
>   and
>   can be selected in a wide range of reasonable values, limited by the charakteristic frequencies of the seismograph system.

> + LOWPASS
>   includes a 4-pole Butterworth low pass ( corner freq. = F-SAMP / 8 ) in the recursive filtering to reduce aliasing errors.

INF and HLP refer to the graphic display windows, where
  informations to the actual cursor position ( INF )
and / or
  hints to actually possible mouse actions ( HLP )
are displayed.

HowTo Table of Content Top of Page

- Screenshots -

SYSTEM : KIRNOS, RESP TO GROUND DISPLACEMENT
( param. see below : screenshot Text Area )

Colors : magenta = SYS_OUT, cyan = SEIS, blue = GALV.

- Frequency Domain -

XFR ( F ) :
Amplitudes ( AMPL ) and Phases ( PHAS ) of the frequency responses

CENTER>

The amplitude response to ground displacement ( SYS_OUT, magenta ) is "flat" for periods from approx. 20 to 0.08 [sec] as a result of the relatively high galvanometer damping
( α = 8.0, see screenshot Text Area ).

Positioning the mouse cursor on the label FCT ( blue, upper left corner )
  shows informations to the curves displayed ( type, scaling and color code, etc. ).

XFR ( Z ) :
Amplitudes ( AMPL ) of the Z-transforms
(F-SAMP 80 [Hz], +LOWPASS )

- Time Domain -

Response to UNIT STEP :
( XFR ( Z ) as above,
red = input = ground displacement [ m, mm, µm ]
SEIS and GALV : normalized, max = 1 )

Response to KUEPPERS_2 :
( param. as XFR ( Z ) above, red = input signal, 100 [Smp] = 1.25 [sec] )

For the input signals KUEPPERS_2 and SWEEP
  the corresponding frequency transform ( red, normalized, max = 100 )
can be included in the display of the frequency response AMPL :

- Text Area -

LIST SEIS :
( KIRNOS, RESP TO VEL, σ^2 = DEF = 0.05 )

The value of Afac(VEL) [s] corresponds to the amplification factor A [s] in
Seismometer-Galvanometer Documentation.

Screenshots HowTo Table of Content Top of Page

- CALEX-Calibration -

- Comments -

The transfer function of the system to be examined ( filter, seismometer and/or galvanometer ) is represented / approximated in the frequency domaine by a product of 1-pol and 2-pol filters.
Characteristic parameters of the product ( amplitude, corner frequencies, damping constants ) are determined in an iteration :

  The response in the time domain of the model filter to an arbitrary input signal is fitted to the response of the system to the same input signal.
  ( topics : least sqare fit, method of conjugate gradients )

  The frequency band of the input signal ( here : sweep ) should cover the the range of characteristic frequency constants of the system / model filter, and
  the input signal and the sytem response should be recorded by the same digital recorder to eliminate its influence.

  Here :
  The applet calculates both time series and feeds them as floating point numbers to a slightly modified JAVA version of the CALEX program ( originally FORTRAN code ), thus avoiding distortions and scaling problems of a recording device.
  Exceptions :
  The geophone parameters of SM4_10Hz and L4_2Hz are determined from highpass filtered input and output signals, to simulate a StrataView recording.

  Corresponding to the original program, input and output are lowpass filtered, to reduce aliasing errors.

The control parameters of the CALEX program, i.e.
  corner frequency and slope of the antialiasing lowpass,
  number, type and order of the partial filters,
  initial values for the parameter iteration ( amplitude, corner frequencies and damping constants )
are set automatically by the applet, depending on the seismograph system selelected, and
are described in detail in the documentation of the FORTRAN prorgram calex.f of E. Wielandt.

The applet lists in its textfield
  the control parameters and initial values of the iteration,
  the iterative modification of the parameter values and
  the final result of the parameter fit
in a format comparable to the original program.

The time series
  input signal "stimulus" ( STIM ORG / +LP ),
  system response ( SYST ORG&nbs;/ +LP ),
  model response ( SYNTH ) and
  difference ( ERROR = SYST+LP - SYNTH )
are displayed in a separate graphic window, updated for every iteration step.

Comments HowTo Table of Content Top of Page

- Program Control -

The CALEX program is controlled by the menue items on yellow background&nb :

The button CALEX ( SWEEP ) activates the CALEX mode of the applet :
  The selection of a seismograph system is disabled ( here previously selected : BENIOFF_LP ),
  the system response is swiched to RESP TO GROUND ACCELERATION and the input function to SWEEP ( def. = DOWN SWEEP, UP SWEEP selectable ).
The frequency range of the sweep signal
  depends on the parameters of the system selected and
  determines the lower limit for the selection of the sampling rate ( F-SAMP ).

Screenshot sweep signal and system response :
( Syst. = BENIOFF_LP, RESP. TO ACC., F-SAMP = 20 [Hz] )

By activating the INIT button
  the options and parameters, and
  resulting control parameters and initial values
are transferred to the CALEX program, listed in text area of the applet, and
  their modification is disabled :

Screenshot text area parameters ( after INIT ) :

TRACE PLOT
  displays the actual state of the iteration in a separate graphic window
  ( see below : screenshot graphic protocol )

RUN / HALT
  starts / stops the iteration,
STEP
  initiates a sigle iteration step.
( After HALT or STEP : RUN => continue )

RESET
  quits the CALEX mode and restores the previous state of the applet.

Screenshot text area iteration protocol :

The lines below CALEX INIT
  list the initial values and ( preceded +- ) the "radii of variation" of the iteration and
  the corresponding amplifications ( GAIN ).

Below CALEX START
  the actual state of the iteration process is displayed, i.e.
  the values of factors applied to the "radii" are listed.

RMS_nrm is the RMS value of the error signal
  ERROR = SYST +LP - SYNTH,
normalized to the RMS value of the system response SYST +LP.

Screenshot graphic protocol :

Program Control Comments HowTo Table of Content Top of Page

- Results -

If the approximation converges or after max. 100 iterations
  the program stops,
  lists the actual results ( true parameter values and dimensions ), and
  displays amplitude and phase of the resulting frequency respons in the corresponding graphic displays of the seismograph system ( see screenshot AMP / PHS ).
( RUN continues the iteration for up to 100 steps )

Screenshot text area results :

The warning between lines 78 and 79 ( UPD SYSTEM ... ) refers to the fact that in step 79 of the iteraton at least once the value 1 for the damping of the galvanometer ( lp2 ) occurred ( not implemented in the the iteration calculus an replaced by 1.0002 ).

In the applet the amplitude ( [m/s^2] ) of the input signal is known, and
the amplification of the system for
  acceleration ( ACC GAIN ),
  velocity ( VEL GAIN ), and
  displacement ( DSP GAIN )
can be calculated from the factor amp.

Screenshots AMPL / PHAS :
( red : normalised spectral amplitude of sweep signal,
green : amplitude / phase of frequency response of the system, calculated from the results of CALEX-program,
and hiding the frequency response from original parameter values )

Results Program Control Comments HowTo Table of Content Top of Page

- Download -

Class and html files for a local installation of the applet are available as a zip file and as a tar.gz file.

More applets at : Geophysics Department TU Clausthal

Rev. 12-nov-2012

Comments to Fritz Keller
( ned gschempfd isch globd gnueg )

Top of Page Table of Content HowTo CALEX Calibration