Measuring the transfer function of a superconducting gravimeter using step functions and sine waves

Michel Van Camp,

Royal Observatory of Belgium, Avenue Circulaire 3, B-1180 Bruxelles, Belgium,

™ michel.vancamp@oma.be, & + 32 2 373 02 65, Ê + 32 2 373 03 39

«

« « « « «

« « « «

Introduction

This note presents a method to determine experimentally the transfer function (amplitude and phase responses) of a superconducting gravimeter by injecting known voltages into the control electronics of the system. Details are given on the analysis of the data by using the Tsoft software. Some parts of this manual summarize papers (see bibliography) as well as numerous personal communications with GWR.

1. Experiments

1.1 Feedback adder circuit

In order to inject a sine or a step voltage in the feedback coil an adder circuit is incorporated into the feedback circuitry. The circuit allows the system transfer function to be characterized in feedback, i.e. in its normal operating configuration. Removal of a jumper also allows characterization of the mechanical system operating out of feedback. This is useful if the user wants to measure he mechanical response of the gravimeter, which depends largely on the spring constant of the levitating magnetic support field; the eddy current damping from the normal conductors around the sphere; and from the He gas surrounding the sphere. Of these parameters the operator can modify only the spring constant. Adjusting the ratio of the upper and lower levitation coils (resulting in a different magnetic gradient) does this. Measuring the mechanical response gives the quality factor Q and eigenfrequency of the sensor (Imanishi et al, 1996) that are important parameters to determine the SG noise due to Brownian motion (see e.g. Van Camp, 1999).

Access to the adder circuit is achieved by detaching the front panel of the GEP-2 electronics and inserting the cable (supplied by GWR with the new GGP card) into the uppermost SMB connector labeled P2. After installing the cable, the front panel can be replaced and loosely secured with one or two screws. This will allow the cable to exit from the chassis between the front panel and the top bezel. The cable is terminated with a BNC connector.

In-feedback (normal configuration)

To operate the circuit, remove jumper J1-B and install the jumper in position J1-C. There is no need to power down the system while changing this jumper. It is easily accessible at the outer card edge after removing the front panel of the GEP-2 electronics. Install the SMB to BNC cable provided into P2 located at the upper corner of the circuit card. Apply a voltage source to the BNC end of the cable. Note that the circuit will not operate correctly if this input is allowed to float while jumper J1C is selected. If a voltage source is not applied the BNC connector should be terminated (shorted) in order for the circuit to operate correctly.

The voltage applied at this point will be added to the normal voltage that comes from the integrator. Then, this summed voltage will be converted to a feedback current through the feedback resistor.

Note: J2, J4 and J1A are not modified.

Out-of-feedback

Change the control switch from RUN to (out-of-feedback) SET-UP. Remove jumper J1B to J1C and in addition remove J1A. Inject the sine or step signal at P2-1. In this case, the output of the integrator remains at zero voltage, so only the injected signal goes to the feedback coil via J1C. The output comes out of Grav Bal (P1-6) on the front panel meter of the GEP-2 electronics. Notice that there is a 1-second filter in front of P1-6 Grav Bal, which should be corrected. If a shorter time constant is needed one can add resistors in parallel to R9 and R10.

The user should decide what gain when operating out-of-feedback. For example, the user could choose a lower gain in order to inject larger signal so that the SNR is higher; or he could adjust the out-of-feedback gain to be approximately the same as the in-feedback gain. This can be done by comparing the integrator voltage Gav Sig to the Grav Bal voltage when one switches from RUN to SET-UP. The Grav Bal voltage can be decreased (increased) by a factor of 3 for each decrease (increase) in Gravity Bridge Drive BD.

Note that the author have not determined this out-of-feedback experiment.

NOTE: If the step function method of determining instrumental response is used, the user should monitor the "GRAV SIG" output from the front panel of the GEP-2 in addition to the output of the GGP filter. This should be done to confirm that overshoot of the feedback circuit is not clipped during the measurement. Depending on the amount of overshoot, the GGP filter output may indicate that no clipping is occurring thereby resulting in an inaccurate measurement of the instrumental response. Monitoring of the "GRAV SIG" signal on a chart recorder is sufficient to insure no clipping is taking place. This is especially important if the instrumental response is to be determined with gravity cards manufactured before revision 2. On circuit cards before revision 2, the integrator circuit has a faster time constant and is subject to greater overshoot and ringing.

1.2 Wave generator

Note: we just give here a general description. It is not possible to give a complete description of the used data acquisition systems and wave generators as those differ from one user to another one. However, GWR DDASS3 data acquisition system users can find a detailed procedure in the GWR Gravity Circuit Card users manual (GWR, 2000).

The injected voltages into the adder are step functions and sine waves. Both should give the same transfer function that results from the sum of several components:

  1. Mechanical response of the gravity meter;
  2. Transfer function of the feedback integrator and differentiator;
  3. Transfer function of analog filters (e.g. Tide or GGP);
  4. Transfer function of the data acquisition systems;
  5. Transfer function of the digital anti-aliasing filters.

Because reaching a precision of 0.01 s for the time delay is quite difficult, especially at long period (T > 500s), it is worthy to compare sine waves and step functions methods in order to check the quality of the results. It allows also one to determine the transfer function of the permanent data acquisition system (some unexpected time delays can be found). Finally, it gives an opportunity to study how the noise affects the different methods.

The reaction of the gravimeter to the injected voltages is measured by digitizing the input voltages and the output voltages of e.g. the Tide, GGP, … filters. The outputs of these different filters are recorded by permanent data acquisition systems. However, for the experiment, an additional one is needed to record the injected voltage.

Great care must be taken at the timing of this recorder and of the wave generator, that requires a GPS or a DCF77 (available in Western Europe) controlled clock. The user must also pay attention to the grounding of the generator and of the data acquisition system in order to avoid loops or floating devices. In our experiments everything was grounded via the adder reference.

Which waves? Which steps?

To search the transfer function in the tidal band with sine waves, we should inject voltages, of which periods equal 12 h or 24h. This is difficult to realize in practise because:

  1. About ten oscillations are necessary for a good signal-to-noise ratio and such an experiment would last several days during which earthquakes, ocean and atmospheric effects would cause perturbations;
  2. Some generators can not produce such long periods;
  3. Too long periods would corrupt data over too many days.

On the other hand, simulation analysis of the response of the Tide and GGP filters indicates that the Tide filter is flat from 1000s and the GGP1, from 500s. In the sine-waves method, the transfer function is extrapolated in the tidal band by averaging the results obtained at 1000 and 2000s (Tide) and at 500, 1000 and 2000s (GGP1).

The amplitude of the steps and sine waves should be about 5-6V peak to peak and must be zeroed in agreement with the gravimeter offset (higher voltage or another zero can be chosen, as long as the gravimeter does not reach its full scale). About 15 steps should be injected, each should last about 5 minutes (GGP filter) and 8 minutes (Tide filter).

2. Data analysis

The author analyzed the data with the interactive Tsoft software (available free of charge on: http://www.astro.oma.be/SEISMO/TSOFT/tsoft.html). Note that this software was successfully compared with the program Etstep (H.G. Wenzel).

2.1 Step functions

If one injects a step voltage into an instrument, it is possible to get its frequency transfer function by calculating the Fourier spectrum of the differentiated step response function. Earth tides (and other drifts) must be removed from the recording before analysis.

Select the channel by highlighting it (cf. § 3.1.1 in the Tsoft manual). Let us suppose we analyse 4 minutes after the step. First make inactive the first points following the 4 minutes (cf. § 3.2.4) (it is not necessary to inactivate all the following data). Then position the curve cursor (red cross) at the beginning of the step and use the <Calculate|Transfer function> command. Two windows appear: "Transfer function (normalised amplitude)" and "Transfer function (group delay)". The first one gives the amplitude response, normalised with the amplitude at the lowest frequency. The second one gives the group delay in second, with the curve cursor as origin time. The delays appearing at frequencies higher than the corner frequency are meaningless. The numerical values may be exported using <File|Export>.

!!! Important remarks:

  1. Analyzing a perfect step (i.e. a direct jump from 0 to, say, 1) produces a fatal error.
  2. The step must be upward. If you want to analyse downward steps, first reverse the channel sign by using <Calculate|Evaluate expression> (cf. § 3.3.2). It is useful as no significant differences were found in the transfer function between upward and downward steps.
  3. Applying a Least Squares Filter (§ 7.2.2) before analysing steps may strongly improve the group delay results. As this filter is symmetric, the user must delay the output stream by half the width of the filter window.

2.2 Sine waves

By fitting both the input (injected sine waves) and the output signals (instrument response) on the function

,

one obtains the phase and amplitude response of the instrument at the period T. P6(t) is a 6th degree polynomial necessary to remove Earth tides and other drifts. Removing Earth tides by subtraction of a synthetic tide gives similar results. The amplitude ratios and the phase differences of the fitted input and output sinusoidal waves provide the instrumental transfer function.

To fit the input channel, highlight this channel then click the <Calculate|Fit Sinus> menu. A dialog box asks the period T of the sine wave. Press <OK> and another box asks the polynomial degree. Press <OK> and the function

is fitted on the selected channel. Select now the output channel that contains the instrument response to the injected sine waves and do a new fit with <Calculate|Fit Sinus>. The results are now in the text output window.

For all fits, the text output window gives the correlation and the standard deviation, which allow easily one to check the fit (e.g., if the chosen period T is the right one). For all fit, the text output gives also the amplitude a, b and the error of the fitted sine and cosine, as well as the amplitude and the phase . Then, the following text looks like:

PROP WITH PREVIOUS:

AMP 123.123 PHA 123.12 s

0.123 0.12

This is the amplitude ratio between the input and the output, the phase difference and their respective error. These errors result from the least squares fit on the output channel (more exactly: on the last fitted channel). Note that this appears after each fit but has only a signification after the second one.

Evaluating the noise effect on sine waves by bootstrapping

As steps are easy to repeat, we can look at the standard deviation to assess the noise effect. For long period sine waves, it is difficult to perform several experiments, so a bootstrap method (Monte-Carlo simulation) can be useful.

This method consists of producing a noise with a spectrum similar to the real one affecting the measurements. This noise is obtained after calculation of the residuals after fitting a function on an instrumental data set. The smoothed power spectrum of the residuals is multiplied by a Gaussian white noise, which gives a spectrum of which inverse Fourier transform produces a colored noise similar to the real one. A mathematical sine is added. The resulting signal is analysed in the same way as the instrumental data. This procedure is repeated 1000 times for a given period. This gives an error distribution function of the amplitude response and the time lag for this period.

Detailed procedure:

  1. Creating the artificial noise
  1. take the output channel that contains the instrument response to a sine wave of period T (it should be a representative recording (more than 1 hour long)).
  2. Create two new channels, one with (use <Calculate|Evaluate expression>) and the other one with (N is the number of point in one day. e.g. 86400 if sampling rate = 1s).
  3. Using <Calculate|Multilinear regression> fit the output on those two channels, using a 6th degree polynomial P(t).
  4. With <Calculate|Evaluate spectrum> calculate the spectrum of the residual of the fit.
  5. Select <Spectrum|Show cycles per day> and calculate the smoothed power spectrum with <Spectrum|Spectral Power Table>. Frequency increment should be 300 and Maximum frequency 30000 cpd.
  6. Save this power spectrum by using <File|Export> in the spectral power window.
  7. The window "Spectral Power" appears. Select then <Calculate|Create colored noise>: a new channel is created, which contains the noise similar to the real one.
  8. To normalise the colored noise with the real one, calculate the spectrum of both and for each spectrum, calculate <Spectrum|Spectral power inside window>. The result appears in the "Text output" window:
  • POWER OF 1: xxxxx

    POWER OF 2: xxxxx

    •

    B. Monte Carlo simulation

    1. Open the file corresponding to an experiment
    2. With <Calculate|Fit Sinus>, evaluate the amplitude of the sine wave (given in the "Text output" window).
    3. <Others|Plug-ins|MC simuls of sin fit>:
      1. Enter amplitude: this is the amplitude of the sine Wave calculated in the previous point B.
      2. Enter period (s): this is the period of the injected wave
      3. Enter # of simuls: this is the number of MC simulations (e.g. 1000)
      4. Enter noise file: this is the file that contains the noise power spectrum (file created in A.6)
      5. Enter factor for noise: this is the ratio between the power spectrum of the real data and the coloured noise one (cf.. A.7)
      6. Set thread at low priority: if you answer yes, the MC simulation shall work more slowly. This could be useful if you want to do something else during the calculation.
    1. A file out.txt is created, containing the noise simulation on the amplitude and phase calculation. <Calculate|Histogram> allows one to calculate a histogram, giving the distribution function. The "Text output" window gives the mean, the standard deviation and the intervals where 50, 95 and 98% of the results lie.

    3. Reported problems

    Tide filter

    With the new gravity board "GGP", it has been observed on C021 and C026 that the Tide output did not (or nearly not) react to the injected step and sine waves but still recorded the real gravity signal (earth tides, etc.).

    During normal operation, the Tide filter board is connected to the integrator output via P1-3 and the front panel gravity switch on RUN. In contrast, the GGP or Grav Sig filters inputs are directly on the "GGP" board and "hard wired" to the integrator output via the connection PG2. When J1B is removed the GGP filter remains connected to the integrator output but the Tide filter is disconnected. Installing J1C connects the output of the adder to both the feedback resistor and the Tide filter. Therefore, during the calibration, the GGP filter looks at the output of the integrator while the Tide filter looks at the output of the adder U8 (i.e. integrator voltage minus injected voltage).

    When a step or sine function is applied to the adder, the integrator responds to null the output of the adder. The observed signal on the Tide filter is the difference between the applied step function and the integrator response. Note that the Tide signal answers to the injected waves at higher frequencies where the integrator (and also the rest of the sensing unit: levitating sphere, etc.) can not respond fast enough to cancel the input.

    This problem can be corrected by cutting out the wire connected Pin D of the gravity board to Pin 1 of the Tide board. Replace it with a connection from Pin L (gravity) to Pin 1 (Tide). Then install Jumper J4: this makes the input wiring of the Tide board identical to the GGP filter.

    4. Acknowledgments

    We are grateful to Richard Warburton and Eric Brinton for their fruitful collaboration during the experiments and the redaction of this manual.

    5. References

    • GWR Instruments, Inc. Gravity circuit card revision 2.1 Users manual. Document revision number 2.1, March 3, 2000.
    • Imanishi, Y., Sato, T. and Asari K. Measurement of mechanical responses of superconducting gravimeters. J. Geodet. Soc. of Japan, 42, 2, 115-117, 1996.
    • Richter, B. and Wenzel, H.-G. Precise instrumental phase lag determination by the step response method. Bull. Inf. Marées Terrestres, 111, 8032-8052, 1991.
    • Tsoft, 2000: A software package for the analysis of Earth Tides and Time Series; manual available free of charge at http://www.astro.oma.be/SEISMO/TSOFT/tsoft.html
    • Van Camp, M. Measuring seismic normal modes with the GWR C021 superconducting gravimeter. Phys. Earth and Planet. Inter. 116, 81-92, 1999.
    • Van Camp, M., Wenzel, H.-G., Schott, P., Vauterin, P. and Francis, O. Accurate transfer function determination for superconducting gravimeters. Geophys. Res. Let., 27, 1, 37-40, 2000.
    • Wenzel, H.-G. Accurate instrumental phase lag determination for feedback gravimeters. Proc. 12th Int. Symp. Earth Tides, 191-198, H.T. Hsu (ed.), Science Press, Beijing, New-York, 1995.