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1 Royal Netherlands Meteorological
Institute (KNMI)
Seismology Division
Wilhelminalaan 10
3732 GK, De
Bilt,
Netherlands
sleeman{at}knmi.nl
(R.S.)
2 University of Utrecht
Faculty of
Geosciences
Budapestlaan 4
3584 CD, Utrecht,
Netherlands
wettum{at}geo.uu.nl
jeannot{at}geo.uu.nl
(A.V.W.,
J.T.)
 |
Abstract |
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 Top Abstract IntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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Introduction |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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Current high-resolution data-acquisition systems are specifically designed to cover a large part of the bandwidth and dynamic range of the sensors. These digitizers are based on delta-sigma modulators (e.g., Candy and Temes, 1992), which decrease the quantization errors at lower frequencies at the price of increased quantization errors at high frequencies. By using a high initial sampling rate (in the order of tens of kilohertz) the quantization error will decrease in the frequency range of seismic interest (e.g., f < 200 Hz). Through the use of these "noise-shaping" digitizers the dynamic range of seismic dataloggers, often expressed by a single number representing the ratio of the largest to the smallest signal that can be recorded (Bennett, 1948), may range up to about 145 dB. The representation of the behavior of a digitizer by a single number is convenient but does not reflect the true dynamic behavior of the digitizer as function of the frequency. First, the dynamic range of the digitizer is not a static value but depends on the sampling rate and the frequency. The noise-shaping effect of the oversampled delta-sigma digitizers results in a larger dynamic range at lower sample frequencies. Second, at lower frequencies (e.g., f < 1 Hz) the self-noise of the digitizers will increase like in any other active electronic component. This so-called 1/f type noise may therefore decrease the dynamic range at lower frequencies. Also, the generation of additional noise over the total frequency band due to nonlinearity or distortion in the system may decrease the dynamic behavior.
The choice of a particular type of sensor and digitizer is usually driven by constraints on the frequency band and the amplitude range of interest. Not only for selection criteria it is important to have this type of information available, but also in the process of data interpretation. For example, the presence of 1/f noise may bias the data analysis and the resolution of the digitizer (in the frequency band of interest) determines the minimum amplitude difference that can be resolved properly. Today's high dynamic range digitizers are specifically designed to match the present generation of seismic sensors. The use and development of other sensors, like superconducting gravimeters (e.g., Freybourger et al., 1997; Rosat et al., 2003; Warburton, 2004) showing lower self- noise than current devices, may put additional demands on the dynamic range and resolution of digitizers. The noise reduction in vertical seismic recordings below a few millihertz with local barometric pressure correction (Roult and Crawford, 2000) permits the achievement of noise levels well below the NLNM (Zürn and Widmer, 1995; Beauduin et al., 1996; Widmer-Schnidrig, 2003), which means that the NLNM may need some minor revision. Also, the analysis technique by Berger et al. (2004) applied to recordings from the Global Seismographic Network (GSN) shows noise levels below the NLNM. The interpretation of such low-noise data would only make sense if the noise level of the data is above the noise levels of sensor and digitizer at these low frequencies (Clinton and Heaton, 2002).
The main purpose of this article is to present and use a robust technique to measure and model digitizer noise and to identify the frequency range in which the digitizer can be used without precaution. As a consequence the method will also reveal under which conditions the interpretation of noise records may be biased by the recording system. The first section describes the relationship between the dynamic range of a digitizer, the number of quantization levels (bits), the clip level, and the sampling rate. This relationship will be useful in modeling the behavior of the digitizer. The technique to compute the frequency-dependent noise level of three-channel digitizers, based on the Modified Noise Power Ratio test (McDonald, 1994), is described in the next section. The method in this article uses three digitizers with a common broadband input. Coherency analysis of the output recordings provides the power of the noise (for each channel) as function of frequency. In the next section this technique is applied to both a Quanterra Q4120 datalogger (www.kinemetrics.com) and a NARS datalogger (www.geo.uu.nl/Research/Seismology/Logger) to reveal the 1/f behavior and the resolution of the digitizers. For both dataloggers a simple model is presented to approximate the power spectral density of the digitizer noise. In the last section we show seismic background noise observations at station HGN (http://www.orfeus-eu.org/working.groups/wg1/station.book/HGN/HGN.html) as recorded by an STS-1 and a STS-2 sensor in the same vault. After correcting the recorded data for the instrument response, the vertical STS-2 recordings have noise levels of 10–15 dB above the STS-1 recordings in the frequency range 0.01–0.001 Hz. Having the Quanterra Q4120 digitizer noise model we can exclude the contribution of the digitizer noise to be responsible for this difference.
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Dynamic Range of a Digitizer |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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| (1) |
within the interval [–/2, /2], where
is the quantization interval, the variance of the error is given by:
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| (2) |
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| (3) |
. The quantization
interval , also called the least-significant bit (LSB), and
the full-scale input amplitude 2A determine the number of quantization
levels
(Oppenheim and Schafer, 1998, p.
205):
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| (4) |
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| (5) |
A more realistic way to specify the dynamic range of a digitizer is as a
function of frequency. This is because an inherent characteristic of digitizers
based on solid-state devices (like semiconductors) or photoelectric devices is
the presence of 1/f type noise, noise whose power spectral density is
inversely proportional to frequency. Also the saturation level of digitizers is,
in general, frequency dependent. In this article, however, we assume that
the clip level remains constant over the frequency band of interest so that the
frequency dependence of the dynamic range is only determined by the quantization
noise. In an ideal digitizer (assuming white quantization noise) the
quantization noise power (equation
2) is uniformly distributed
between dc and the Nyquist frequency fN
(Aki and Richards, 1980, pp.
597– 599). Because the noise power is independent of the sampling rate
(Fig. 1), the (one-sided) power
spectral density, PSDnoise(f), of the
quantization process is:
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| (6) |
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| (7) |
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Dynamic Range Measurement |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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There are several ways to calculate and represent the dynamic range of a
system (Hutt, 1990). One way is
to calculate the ratio of the maximum peak amplitude of the recorded self-noise
and the clip level. A more common way is to measure the dynamic range in a
specified frequency band, and express this by a single number as the ratio of
the root mean square (rms) of the noise and the rms of a full- scale sine wave:
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| (8) |
The PSDs shown in this article are estimated by using Welch's averaged periodogram method (Welch, 1967). In this method a time series is divided into a number of overlapping sections of data. In each section we remove the mean, apply a taper on the data, and calculate the power spectrum by taking the square of the discrete Fourier transform of the tapered data. Finally, an estimate of the PSD is obtained by averaging the power spectra over the number of sections. The windows are tapered with a normalized Hanning window and have an overlap of 50%. All PSDs are calculated by using the definition from engineering where the power is attributed to positive frequencies only. This so- called one-sided PSD (as opposed to two-sided PSD in which power is attributed to positive and negative frequencies [Press et al., 1988, pp. 401–402]) was used by Peterson (1993) to construct the NLNM. Our results are compared with the NLNM.
The short-circuited input test does not measure the nonlinearity or distortion of the system. On the other hand, delta- sigma modulators may produce periodic oscillations when there is no input signal (Baker, 1997). One way to reduce this idle tone problem is to introduce a small offset voltage to the input signal. Our results do not show idle tones in the short-circuited input tests. Nonlinear behavior of delta- sigma modulator electronics may be introduced with large- amplitude input signals and generate additional noise that depends on the signal level. This additional signal-generated noise may therefore decrease the dynamic range at large- input signals. The effect of this type of generated noise can be measured by feeding multiple digitizers with a common, large, and well-defined signal. Historically, the linearity of a seismic system has been measured in a two-tone test (Steim, 1986), in which the system is driven by two sine waves with nearly identical frequencies (Hutt, 1990). In this report, however, we focus on the behavior of the self-noise only, in the presence of a common input signal. In this report the common input for the digitizers was the output of the vertical component of a STS-2 sensor. The sensor was located at a site at which the ratio between the seismic background noise and the estimated digitizer noise was roughly about 40 dB over a large frequency range. After removing the coherent signal between the digitizer outputs, each digitizer output will reflect its self-noise.
To estimate the self-noise of digitizers we developed a new technique using coherency analysis. In the conventional approach to estimate the self-noise of linear systems, two systems are used and fed by a common, coherent input signal. This technique has been used in many studies to calibrate seismometers (e.g., Berger et al., 1979; Holcomb, 1989; Pavlis and Vernon, 1994), in which two seismometers are placed close together so that it can be assumed that they record the same ground motion. The mathematical solution of such a system is very simple, but the practical application is limited because the method assumes that one of the pairs of sensors has an accurate known frequency response. Small errors in the transfer functions (or gains) in the two linear systems will cause relatively large errors in the calculated noise levels (Holcomb, 1989). Our approach uses three linear systems that are also fed by a common input signal. The following mathematical description of the model shows the advantages of this approach as opposed to the conventional two-channel approach.
The output yi of digitizer i can be written as
the convolution of the input signal x with the digitizer's impulse
response hi, plus the internal noise ni:
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| (9) |
denotes convolution. Some two-
channel models add noise to the input signal
(Holcomb, 1990), but for the
purpose of self-noise it is appropriate to add transfer-function-independent
noise. For analog systems there is no difference in this approach as the
relation between the noise at the output and the noise at the input is defined
by the (well enough known) transfer function. For digitizers one can expect
slightly different statistical properties between the quantized output noise and
the (analog) input noise. Equation
(9) translates in the frequency
domain to:
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| (10) |
|
| (11) |
j the noise cross-power spectra Nij is assumed to
be zero, so that:
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| (12) |
j k. This equation reveals that the ratio between the
transfer functions of digitizer j and k can be estimated
solely by the ratio of the cross-power spectra between channels j and
i and the cross-power spectra between channels k and
i. Taking the ratio between autospectra Pii and
cross-spectra Pji gives:
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| (13) |
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| (14) |
j k. This equation expresses the power spectra of the
system noise, only in terms of the cross-power spectra and autopower spectra of
the recordings of the three digitizers connected to the same (analog) input
signal. The mathematical description of the three-channel linear system model
shows that we can estimate, solely from the output recordings, (1) the ratio of
the transfer functions between the channels and (2) the noise spectrum for each
channel. We do not need to know the transfer functions, or its accuracy as is
required in the two-channel model. |
Tests and Results |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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To extract the dynamic range as a function of frequency the short-circuited time series are processed using equation (14) on recordings of 2 hr for the 100 samples/sec data streams and 4 hr for 20 samples/sec data (Fig. 2). For frequencies above (roughly) 1 Hz the noise is flat and the PSD does not vary significantly with frequency. However below 1 Hz the 1/f type of noise dominates and the dynamic range decreases at lower frequencies. The horizontal lines show theoretical PSD levels for 22-, 23-, 24-, and 25-bit digitizers as derived from equation (7). The corresponding values for the dynamic range of the digitizers are given on the right axes and follow from equation (5). The results in Figure 2 show that for frequencies above (roughly) 1 Hz the dynamic range for the three digitizers corresponds within a few decibels with the values in Table 1.
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The effect of using a real seismic broadband signal on the dynamic range is shown in Figure 3. The digitizer outputs (recordings from the vertical component of an STS-2 sensor) are processed by using equation (14) and the same window lengths as above. For the 100 samples/sec and 20 samples/sec data the noise PSD of the shorted input measurement and the common input test are shown in gray and black lines. Notice that only one digitizer is shown to see the difference. At both sampling rates there is significant increase in the self-noise level and hence no significant decrease of the dynamic range. Figure 4 shows the measured gain ratios between the digitizers in the Q4120 datalogger, smoothed over a tenth of a decade. Between 0.01 Hz and 8 Hz, using data sampled with 20 samples/sec, the smoothed ratios are within 1.6% (or 0.14 dB) of the values given by the manufacturer.
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These procedures are also applied to a NARS datalogger. The result of the coherency analysis on 20 samples/sec data is presented for one digitizer in Figure 5. Over the entire frequency range (0.001–8 Hz) some additional self-noise is visible, which decreases the dynamic range by a few decibels. At higher frequencies (1–8 Hz) the dynamic range increases to about 127 dB at 20 samples/sec, corresponding to a 20.8 bits digitizer. The PSD level at lower frequencies shows a significant smaller slope as compared with the Q4120. For frequencies below 0.01 Hz the PSD level of the NARS datalogger is below the Q4120 PSD level.
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Digitizer Noise Models |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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The behavior of the Q4120 digitizer in
Figure 3 at 20 samples/sec can
be modeled by the superposition of white noise dominating at high frequencies
and 1/f type of noise dominating at low frequencies
(Fig. 6). The model in this
figure is based on equation (7)
and represents a digitizer with a 23.6-bit broadband spectrum and a 24.7-bit
spectrum with 1/f1.55 noise:
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| (15) |
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The behavior of the NARS datalogger noise is modeled in
Figure 7 by the same type of
superposition as in equation
(15): a 20.8-bit broadband
spectrum dominating at high frequencies and a 23.0-bit spectrum with pink noise
(1/f1.0) dominating at low frequencies:
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| (16) |
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Discussion |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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However, STS-1 and STS-2 sensors are not noise-free devices and, in fact, their self-noise is comparable to the Quanterra digitizer. It may therefore be difficult to separate the sensor noise and the digitizer noise in background noise recordings. As an example (Fig. 9) we have analyzed data from two systems at station HGN (Heimansgroeve, Netherlands): a shielded STS-2 (third generation; G. Streckeisen, personal comm., 2001) connected to a Q4120, and a STS-1 connected to a gain-ranged acquisition system (Dost and Haak, 2002). Both systems are in the same vault at a distance of only 2 m. Data for the same period (2002) are deconvolved with the instrument response and shown in Figure 9 for three components. The horizontal noise (both components) is about 30–50 dB above the NLNM at frequencies below 0.01 Hz and is equally recorded by both systems. On the vertical component, however, a significant difference is visible below approximately 0.01 Hz. The background noise recorded by the STS-1 closely follows the NLNM, but the STS-2 recorded noise is 10–15 dB higher. Different noise sources may contribute to this difference, for example, barometric pressure fluctuations, variation in temperature, sensor noise, and digitizer noise. However, the effect of ground tilt and elastic response of the Earth due to atmospheric pressure fluctuations would be the same for the two instruments because they are on the same pier. This is confirmed by the similar, observed horizontal noise levels recorded by both instruments. A periodic deformation (at 1.7 x 10–3 Hz) of ±1 µm over a distance of 3 km, would result in horizontal noise which is about 40 dB above the vertical noise (Wielandt, 2002b). Local pressure variations in the vault may affect the seismometers in at least three ways (Wielandt, 2002b): (1) a buoyancy force when the sensor is not sealed, (2) adiabatic changes of temperature, and (3) deformation of the sensor housing. However, the casings of the STS-1 and STS-2 sensors are designed to suppress these kind of effects. Also the sensors are thermally insulated; the STS-2 is covered with fiber wool and a heat-reflecting blanket, all covered by a stainless steel jacket, and the STS-1 instrument is covered by a partially evacuated glass bell. Consequently, we assume that the temperature effect for the frequencies between 0.001 and 0.01 Hz does not have a great impact on the noise level. Another reason for the discrepancy between the STS-1 and STS-2 vertical-component observations in Figure 9 could be related to deviations between the measured and the manufacturers "nominal" response. Fels and Berger (1994) measured response deviations for a STS-1 of up to 1% in amplitude and 1° in phase between 0.2 and 100 mHz, and up to 12% in amplitude and 5° in phase for frequencies above 5 Hz. These deviations, however, are too small to explain the difference of 10–15 dB. This is also confirmed by the observation that the horizontal components do not show such a significant difference.
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The previously modeled digitizer noise (equation 15) is useful here because it excludes the digitizer as origin of the observed difference in noise level. Between 0.01 and 0.001 Hz the digitizer model is about 10 dB lower than the NLNM and could therefore not contribute to this difference. Given these observations and assumptions the discrepancy in noise level seems to be caused by the sensor noise level, although this seems in contradiction with the noise level presented in the STS-2 manual (Streckeisen, 1991). To validate this explanation we compared the observations with the noise model of the STS-2 by Wielandt and Widmer-Schnidrig (2002) in Figures 10 and 11. Figure 10 compares the NLNM, the STS-2 noise model, and the digitizer model. Figure 11 compares the STS-2 noise model with the STS-1 and STS- 2 observations at HGN. The noise model fits very well with the STS-2 observations because it marks within a few decibels the lower boundary of the STS-2 observations. Also the slope of the 1/f noise model fits very well the observations. However, recent studies by Widmer-Schnidrig (2003) and Berger et al. (2004), respectively, show smaller (5–6 dB) and larger (19–24 dB) differences between vertical STS-1 and STS-2 noise levels for frequencies between 1 and 2 mHz. Several factors may contribute to the differences, such as the quality of installation of seismometers, the variable quality among the different STS-2 sensors, or differences in self-noise between STS-2 sensors. Our new technique will be ideal to measure the self-noise of STS-2 (and other) sensors by placing three sensors together at the same location. Differences in quality between sensors and self-noise, or inaccurate knowledge of transfer functions, will not bias the measurement of the self-noise.
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Conclusions |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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Acknowledgments |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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Manuscript received February 23, 2005
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References |
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TopAbstractIntroductionDynamic Range of a...Dynamic Range MeasurementTests and ResultsDigitizer Noise ModelsDiscussionConclusionsAcknowledgmentsReferences |
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