Transcript (minimum-delay or zero

International Training Course, Rabat 2012
Title
E. Wielandt:
Relative (electrical) calibration of seismometers
Systematics
Recorded?
Assumed?
Text
Relative calibration
(determining the frequency response)
• A variety of signals - sinewaves, pulses, pseudo-random signals - can
be used for seismometer calibration. However, it is risky to rely on the
precision of a waveform generator. Preferably, the input signal should
be recorded together with the output signal.
• The transfer function is defined as the complex gain for sinewaves,
but in order to make practical use of this relationship, one must wait
for a steady state after each change of the frequency. This is not
practical for broadband seismometers. Also, we want a mathematical
representation of the response, not a table for discrete frequenies.
• Modelling the output signal from the input signal (preferably in time
domain), and matching the parameters of the mathematical model, is
a more efficient method of determining the frequency response.
Simplest but not best: measuring the amplitude
response of a geophone to sinewaves
By superimposing standardized response curves,
the parameters can be directly determined
( 4.5 Hz geophone with 0.4 of crit. damping)
Text
Resonance curves can of course be fitted automatically,
and even the exact frequency of the sinewaves can be
determined from the record. The only requirement is
that the input sinewaves all have the same amplitude.
This method is quite useful for the remote calibration
of geophone-type sensors.
The next slide shows the output signal from which the constants
of a Geotech S13 in the GERESS array were determined with
the SINCAL routine (download) as:.
T0 = 0.979 s, h = 0.774
sincal
SINCAL2
Period
Damping
Gain
best fit
0.932
0.7481
1503.96
boots.avg.
std.dev.
0.932 +0.000
0.7479 +- 0.0006
1503.84 +0.68
halfbridge
If the geophone has no calibration coil, then the
signal coil can be used for both input and output. The
voltage across the coil caused by the input current
can later be automatically subtracted from the output
(program CALEX3, download).
10 Hz geophone
Calibration of a 10 Hz field geophone with a half-bridge, using a
square wave as an input. From top: input current, recorded output
voltage, modeled output voltage, misfit. Note that the misfit is not
the same for the mass jumping „up“ and „down“, indicating a
dependence of the free period on mass position, i.e. a nonlinearity.
Such problems can only be recognized in time domain.
cal. with arbitrary signals
Hardware used for calibration with a logarithmic sweep. The left box
splits the signal into adjustable U, V, W components for the STS2,
and can deliver purely „horizontal“ or „vertical“ calibration signals.
It is also possible
to calibrate the U, V, and W sensors separately - the Z
output may be used for this purpose - and then to average
the U, V, W transfer functions or parameters with a matrix
whose elements are the squares of those of the matrix
transforming the U, V, W into the X, Y, Z components:
a sweep calibration
Calibration of an STS2 seismometer with a sweep signal from
20 s to 400 s period. The rms residual is only 1/3000 of the rms
output signal, and consists mainly of ambient long-period noise.
A slightly nonlinear seismometer (diode circuit added)
another sweep calibration
Another example. The sweep this time covers a band from 20 s to
800 s. The relatively large and strange-looking residual is caused by a
defective 16-bit digitizer that produces a small jump in every zero
crossing. Time-domain modelling with the CALEX method helps to
identify such problems.
Calibration of Digitizers
Digitizers normally don’t need to be calibrated if the manufacturer’s
specifications are clear and complete.
You may want to check the scale factor, normally in microvolts per count, by
connecting a battery and a digital voltmeter to the input.
Forget about the filter coefficients! You only need to know the filter type
(minimum-delay or zero-phase).
For all frequencies lower than one-quarter of the sampling rate, (that is, one-half
of the Nyquist bandwith) you may assume that the response is flat and the phase:
- of a zero-phase filter is zero
- of a minimum-delay filter represents a constant delay whose magnitude you can
experimentally determine by recording a time signal (such as from the pps output
of a GPS receiver). At low frequencies, even a minimum-delay filter is nearly
linear-phase (but not zero-phase).
- You should also check if the delay of a zero-phase filter is really zero. There
might be an error in the time-tag.
„Minimum phase“ is normally maximum phase
The term „minimum phase“ was coined with respect to a
Fourier transformation with exp(jwt) in the forward
transformation. Most seismologists use however exp(-jwt).
Then the signal with the smallest possible delay – the
minimum-delay signal – has, mathematically speaking, the
largest possible (although negative) phase.
The unambiguous term „minimum-delay“ is preferable to
the term „minimum-phase“.