A reference acoustic sensor with well-characterized complex frequency response from DC to several hertz would enable comparison-calibration of transducers and arrays designed for detection, localization, and characterization of infrasonic acoustic events. That frequency range is challenging for established calibration techniques. One conventional approach is to use an infrasound measurement microphone as a comparison-calibration reference; however, as frequency decreases below a few hertz, the uncertainty in the microphone response typically increases. In a comparison calibration, that uncertainty contributes directly to the uncertainty in response of the sensor under test. An alternate approach, described in this paper, uses a “barometric” sensor—a sensor with known response at zero frequency—as a comparison-calibration reference. The goal is to bridge the region from DC to tens of hertz with a low-pass model for the complex frequency response of the barometric reference while preserving the low uncertainty of that reference. In this way, a laboratory lacking primary-calibration capability could perform traceable comparison calibrations with low uncertainty in the hundredths to tens of hertz region.

Although technically perceivable at high-pressure levels even for frequencies below 1 Hz,1 infrasound is usually defined as the “inaudible” acoustic frequency band below 20 Hz. The combined effects of low attenuation2 and waveguides that result from temperature and wind gradients in the Earth's atmosphere3 result in low-loss transmission. Consequently, many large infrasound sources can be detected at long ranges by large-scale networks of arrays making infrasound a practical means for observing both natural and man-made energetic phenomena4 such as volcanic eruptions, large chemical explosions, and even tornados.5 

The frequency band of interest for infrasound stations in the International Monitoring System (IMS) of the Preparatory Commission for the Comprehensive Nuclear-Test-Ban Treaty Organization extends from 0.02 to 4 Hz6 —a challenging range for established acoustical calibration.7 Barometric-pressure sensors are intended for measuring static (DC) pressure; however, they often have usable response8 into the infrasound frequency range. If a suitable model for the frequency-dependent response of a DC pressure sensor is available and if that model depends only on its response at DC and one or more easily determined parameters, this model-based approach would permit using the DC pressure sensor as a traceable comparison-calibration reference for infrasound. The requirements are (1) a model for the complex response (magnitude alone is insufficient) as a function of frequency and (2) knowledge of the uncertainties in the model parameters. Recent measurements suggest that these requirements can be met with a particular DC pressure sensor for frequencies from zero to tens of hertz. This paper focuses on the Setra 278 “barometric-pressure sensor”; however, the technique is not restricted to that sensor. It is only necessary that the DC sensor be well described by a complex, frequency-dependent model, and to have roughly flat response from DC to at least 10 Hz.

In the context of calibration, the distinction between the terms “static” and “dynamic” (or “DC” and “frequency-dependent”) is artificial. Strictly, static response is the dynamic response in the limit as the frequency goes to zero. Dynamic response is often complex with explicit frequency dependence while static response is real with no explicit frequency dependence; however, these characteristics of static response are natural results of letting frequency approach zero. In this work, the terms static and DC refer to the dynamic quantity in the limit of zero frequency. In this respect, there is no conflict in developing a model to bridge a gap from “zero” frequency to some non-zero frequency.

In calibration, it is common to rely on one or more physical models, the parameters of which may depend, for example, on static values of pressure and temperature. In ordinary coupler reciprocity9 for condenser measurement microphones, the transfer admittance—a complex, frequency-dependent quantity—is, at minimum, a function of coupler volume, the ratio of specific heats, and DC pressure: real-valued, “static” quantities. The coupler admittance model allows calculation of response at frequencies well above zero even though the model is based on DC measurements of pressure and temperature. Such model-based calibration is recognized and described in Part 6 of the Guide to the Expression of Uncertainty in Measurement,10 maintained by the Bureau International des Poids et Mesures (BIPM).

This paper describes the adaptation of a commercial barometric-pressure sensor—the Setra 278, which has a simple low-pass response model—for these purposes. (Specific commercial products are mentioned in this paper to facilitate reproduction of results; however, no endorsement is implied.) The transduction model for the Setra requires only two parameters, the DC response and a characteristic frequency. The DC response and its uncertainty are determined from the sensor's NIST-traceable calibration. The characteristic frequency and its uncertainty can be determined by comparison between the phase responses of a calibrated infrasound sensor and the Setra in the one to tens of hertz region.

While any non-DC infrasound sensor with known complex response would have potential for determining the characteristic frequency of the barometric sensor, this discussion will assume a calibrated infrasound measurement microphone. Standard coupler reciprocity for condenser measurement microphones has been demonstrated at several national measurement institutes (NMIs) down to 2 Hz.11 The French NMI, Laboratoire National de métrologie et d'Essais (LNE) demonstrated magnitude and phase measurement down to 0.01 Hz in a laser pistonphone12 and, in a joint venture, Sandia National Laboratories (SNL) and Penn State University (PSU) demonstrated magnitude and phase measurement down to 0.05 Hz with a large-chamber reciprocity apparatus. This chamber was built by the National Center for Physical Acoustics (NCPA) at the University of Mississippi with subsequent modifications by SNL.

The LNE and SNL/PSU demonstrations were part of an international calibration exercise, PTSAVH.A-C1,13 coordinated by the Preparatory Commission for the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO PrepCom). While these primary-calibration methods are necessary to support standards development, introduction of a suitable very-low-frequency reference sensor for secondary (comparison) calibration would enable traceable infrasound measurements at laboratories not equipped for primary calibration.

For the following discussion, we assume the availability of a DC or barometric-pressure sensor with traceable calibration at zero frequency, and an infrasound sensor with known response above 1 Hz. The goal is to bridge the region from “zero” to several hertz with a frequency-response model for the barometric-pressure sensor so that a laboratory lacking primary-calibration capability could perform comparison (secondary) calibrations with low uncertainty in the hundredths to tens of hertz region.

In this method, the infrasound sensor—in this case, an infrasound microphone—is not used as a transfer standard for conventional comparison calibration; it is used only to find the DC pressure sensor's characteristic frequency. There are three advantages of this approach: (1) because the pressure sensor's characteristic frequency is in the tens of hertz, it is not necessary to know the response of the microphone below 1 Hz; (2) only the phase response is required for the microphone, not the magnitude response; and (3) the low sensitivity of the combined (or total) uncertainty for the DC sensor to uncertainty in the characteristic frequency results in a combined uncertainty well below the uncertainty in the microphone's response.

This paper is a follow-up to work performed by one of the authors (C.L.T., see the supplementary material) as part of his contribution to the PTSAVH.A-C1 multi-national calibration exercise. The present work offers important improvements over the original study including improved traceability and assessment of uncertainty.

The Setra 278 barometric-pressure sensor is one of the three varieties of infrasound sensor evaluated in the multi-national calibration exercise, designated PTSAVH.A-C1, and coordinated by CTBTO PrepCom. In this exercise, two Setra 278 pressure sensors, two Brüel and Kjaer (BK) 4193 infrasound microphones, and two Martec MB2005 microbarometers were calibrated from 0.01 to 10 Hz. Here, the word calibration means determination of complex response—magnitude and phase—in volts per pascal as a function of frequency and the associated uncertainty. Further refinement requires traceability to a national metrology institute (NMI) like the National Institute of Science and Technology (NIST) in the U.S., the LNE in France, the National Physical Laboratory (NPL) in the UK, or other NMIs.

Although the self-noise of the Setra 278 is too high to qualify the Setra for use in the PrepCom's global network of infrasound monitoring stations, the Setra has potential as a low-uncertainty reference for secondary calibration of other infrasound sensors. The PTSAVH.A-C1 exercise and additional work by one author (C.L.T.) demonstrated that a simple second-order-in-frequency model fits the Setra's complex response, S(f), well from zero to tens of hertz,
(1)

The model has only two free parameters, (1) the DC response, S0, a NIST-traceable value (with its uncertainty) provided by the manufacturer and (2) a characteristic frequency, f0. While the manufacturer has not confirmed the validity of this model, measurement by the National Center for Physical Acoustics (NCPA) of the Setra response with respect to a stable, well-characterized microbarometer—a special version of the NCPA U-Series sensor related to the Hyperion Series 5000 sensor—supports this model. Figure 1 shows the results of a fit to the phase response that yielded a value of 70.1 Hz for f0. [For confirmation of the validity of the model, a fit on the magnitude response (not shown) produced a value of 70.1 Hz for f0.]

FIG. 1.

Upper plot: phase response measurement (dots) of Setra 278 SN4744909 from comparison calibration with NCPA U101. The second-order fit (solid curve) is based solely on the measured phase, ϕ. Lower plot: magnitude response (solid curve) using the f0 value from the phase fit and the DC response, S0, to show the consistency of the model with the measured response magnitude, |S| (dots).

FIG. 1.

Upper plot: phase response measurement (dots) of Setra 278 SN4744909 from comparison calibration with NCPA U101. The second-order fit (solid curve) is based solely on the measured phase, ϕ. Lower plot: magnitude response (solid curve) using the f0 value from the phase fit and the DC response, S0, to show the consistency of the model with the measured response magnitude, |S| (dots).

Close modal

In previous work performed by C.L.T. (see the supplementary material), the variation in amplitude with frequency was fit to Eq. (1) using the value of S0 computed from the NIST traceable calibrations performed by Setra Systems. It was pointed out14 that this method broke traceability because the ISO 80000-4:2019 standard15 distinguishes “static” quantities, which do not have explicit time dependence, from “dynamic” quantities. Reference 5 on model-based calibration resolves this apparent conflict. The discussion by Hjelmgren16 further describes the treatment of these quantities.

As discussed above, in this work, we fit to the phase with frequency rather than the amplitude to obtain the model parameter f0. The relative variation in amplitude (“frequency response”) is then used to verify the model fit of f0. We use only the static value of S0 obtained from high accuracy DC calibrations, when computing the frequency response of the sensor, and the (1+jf/f02)1 term is used to extend this static value to “dynamic” (f > 0) values of the sensor response. As long as f0 is computed in a traceable manner, this method can be considered fully traceable for the infrasound frequency range, bridging the gap between zero-frequency barometric pressure calibration and low-frequency infrasound-sensor calibration. The fact that this can be done with a relatively low-cost, off-the-shelf instrument is a key feature of the method presented here.

Fitting to phase rather than amplitude can also yield improved accuracy in the estimates of f0, because the phase, ϕ = tan−1(y/x), is a function only of the ratio of the real, x, and imaginary, y, components of the sensor response; whereas, the amplitude, A, (A2 = x2 + y2) is the quadrature (“squared”) sum of the real and imaginary components. Since ϕ depends only on the ratio of x and y, any multiplicative systematic error is explicitly divided out in the computation of phase. Common examples of multiplicative errors are simple gain errors and non-flat frequency response of the sensor. Details of the method for fitting to f0 using the phase of the sensor response are given in the  Appendix.

For the Setra 278 used in the present calibration demonstration, the manufacturer's data yield a value of 0.1001 mV/Pa for S0 with a k = 2 uncertainty of 0.08%. In this paper, uncertainties are given as standard uncertainty (“one sigma”) times a coverage factor, k, of 2. In practice, this is not significantly different from the 95% confidence interval. The notation (k = 2) is used in this paper as a reminder that the associated uncertainty is approximately two standard deviations.

A reliable method for determining f0 and its uncertainty would enable use of the Setra 278 as a comparison-calibration reference from zero to tens of hertz with sub-percent uncertainty.

Interlaboratory comparisons of barometric calibrations at atmospheric pressures, such as that reported in Bojkowski,17 suggest that primary-laboratory expanded (k = 2) uncertainties as low as 0.005% are attainable. By comparison, the state-of-the-art techniques for 1/2-inch condenser microphones18 are only able to achieve expanded uncertainties to ∼0.3% even at acoustic frequencies. As a result, a traceable method that extends the static pressure calibration of a barometer to infrasound and low frequency acoustic regions could provide greater precision than more traditional technologies such as direct comparison calibration against an infrasound condenser microphone.

It is worth noting that, because of the very different uncertainties in instrumentation for the barometric-sensor based calibration as opposed to conventional comparison calibration using an infrasound microphone, degrees-of-equivalence analyses,19 as used to compare the performance of different laboratories, should be viewed with caution.

From measurements at NCPA of several Setra 278 sensors, the characteristic frequencies, f0, were in the decade from 10 to 100 Hz (see the supplemental information for more details). The DC response is traceable to NIST. With an infrasound microphone (or other infrasound sensor) having known response in the 1 to 100 Hz range, the upper-frequency response of the Setra can be established by determining a value for f0.

Infrasound microphones (e.g., the Brüel & Kjaer BK4193 or the GRAS 46AN/AZ) are available that have relatively flat response from several hertz to several kilohertz. Any sensor well-characterized in the 10 to 100 Hz range should be acceptable. The response does not have to be flat but must be known. Condenser measurement microphones are cited here because they are readily available and calibrations have been demonstrated to below one hertz. If the response and uncertainty of one of these microphones is known (by LNE laser-pistonphone calibration, for example), then a least squares fit of the Setra model function to a phase-response comparison with the infrasound microphone would permit estimation of f0 and its associated uncertainty.

As discussed below, the combined (or total or overall) uncertainty in a sensor's frequency response depends on the uncertainties of components of the sensing system and on the sensitivity of the effect of each component on the combined uncertainty. In a standard comparison calibration against an infrasound reference sensor, the combined uncertainty normally grows with decreasing frequency. The method described here ties the low-frequency result to the response of a barometric-pressure reference, which can yield smaller combined uncertainties than for the standard comparison method.

The general uncertainty-propagation relationship [e.g., Ref. 20, Sec. 5.1.2, Eq. (10)] expresses the combined uncertainty, uG, of a function, G, where G is a function of N components, xi, as
(2)
if the variations in the components are uncorrelated. For uncorrelated components, the combined uncertainty, uG, is the square root of the sum of the squares of the component uncertainties, u(xi), weighted by the component sensitivities. The equivalent expression for correlated components would contain, in addition to the squares of the components, cross-products between components weighted by component sensitivities and by cross correlation coefficients.

The u's are the standard uncertainties of the components, xi, and uG is the combined standard uncertainty of the function, G. (If the distributions of the components are normal, the standard uncertainty becomes the standard deviation.) The partial derivatives are factors that indicate how sensitive the combined uncertainty is to the component uncertainties. Notice that the u's have units—the same units as the corresponding xi—and that uG has the same units as G. Keeping track of the units can prevent errors in application of formulas.

Often, fractional uncertainties—non-dimensional quantities—are more simple to interpret. The fractional uncertainty follows from dividing Eq. (2) by G2 and then grouping factors on the right side to expose the fractional uncertainties of the components,
(3)
The quantity in square brackets on the left is the combined fractional uncertainty in G; the quantities in square brackets on the right are the fractional uncertainties of the components, xi; and the quantities in large parentheses on the right are the sensitivities of the combined uncertainty to the uncertainties in xi.

In this paper, the desired result is either the combined uncertainty of the response magnitude or the combined uncertainty of the response phase and the components of uncertainty are the parameters, S0 and f0, in the response equation, Eq. (1). There are three expressions of component uncertainty:

  • standard uncertainty, u(xi), the standard deviation of xi (if normally distributed) with the same units as xi;

  • fractional uncertainty, u(xi)/xi (unitless); and

  • percentage uncertainty, which is 100 times the fractional uncertainty (also unitless).

The combined uncertainty, uG, has the same three options for expression.

As discussed previously, a second-order low-pass function, S(f), with two parameters, S0 and f0, fits the Setra 278 well up to several tens of hertz [see Eq. (1)]. For the Setra, variations in S0 derive primarily from the physical properties of the sense element while variations in f0 are associated with the internal electronics that follow the sense element; consequently, the assumption of zero correlation between S0 and f0 is reasonable and Eqs. (2) and (3) are appropriate.

From the point of view of uncertainty, we will consider separately the magnitude, |S|, and the phase, ϕ, of S, along with the components of uncertainty of S0 and f0. Note that, in this paper, we focus on the uncertainties associated with the barometric sensor, the Setra 278. A complete calibration would also consider uncertainties associated with conditioning electronics and data-acquisition systems.

To evaluate the uncertainty in magnitude, let the G function in Eq. (3) be the response magnitude,
(4)
which depends on two parameters (the “components”), x1 = S0 and x2 = f0. S0 and f0 have separate fractional uncertainties (the “component uncertainties”), uS0/S0 and uf0/f0, and each component has a sensitivity factor that weights the component uncertainty in the propagation equation, Eq. (3). The square of the combined fractional uncertainty for the magnitude is then
(5)
where the fractional uncertainties—combined on the left, components on the right—are in square brackets and the uncertainty sensitivity factors are in large parentheses. Note that the individual terms in Eq. (5) are non-dimensional and that percentage uncertainty is 100 times fractional uncertainty. The partial derivatives applied to Eq. (4) give direct expressions for the sensitivity factors
(6)
and
(7)
consequently,
(8)

The second sensitivity factor, Eq. (7), illustrates an important aspect of maintaining low-uncertainty in the Setra response function. As described in Sec. II, f0 is typically greater than 10 Hz. With f0 at the low end of the range (10 Hz), the sensitivity factor would be 1.0 at 10 Hz, 0.4 at 5 Hz, and about 0.02 at 1 Hz. For frequencies below f0, only a fraction of the uncertainty in f0 contributes to the combined uncertainty of the response magnitude and that fraction decreases as the frequency decreases.

For the phase of S(f), the G quantity is proportional to the arc-tangent,21 
(9)
and there is only one parameter, x1 = f0. Since the S0 parameter and its uncertainty are given by the manufacturer, the phase approach is sufficient (and preferable) for defining the response model. Since radian phase is already a fractional quantity (arc length per radius), we use Eq. (2) with the appropriate substitutions,
(10)
The partial derivative of ϕ with respect to f0 is
(11)
therefore,
(12)
which expresses the uncertainty in radian phase in terms of the fractional uncertainty in f0 (square brackets) times the uncertainty sensitivity of the component (large parentheses), f0.

As for the magnitude, for frequencies below f0, only a fraction of the uncertainty in f0 contributes to the combined uncertainty of the response phase. Notice that the uncertainty sensitivity of the component, f0, in the magnitude expression [Eq. (7)] is roughly quadratic in frequency while the uncertainty sensitivity of f0 in the phase expression [Eq. (11)] is roughly linear. In both cases, the uncertainty sensitivity drops as the frequency decreases below f0; consequently, the combined uncertainty also drops with the caveat that we are ignoring contributions to uncertainty from, for example, signal-conditioning electronics or noise.

In order to use the DC pressure sensor—the Setra, in this case—as a reference sensor for comparison calibration, the two parameters for its response model and the associated uncertainties must be known. The first parameter, the DC response, S0, and its uncertainty are given from data on the manufacturer-supplied calibration sheet (or given by DC pressure calibration). The second parameter is the characteristic frequency, f0.

One approach for estimation of f0 is to compare the Setra output with the output of a known reference. In this example, the known reference is a BK4193 (“B&K”) infrasound microphone calibrated by LNE. This estimation was done for Setra 278 SN4744909 in Sec. II with the result that f0 is 70.1 Hz by least squares phase fitting (see the  Appendix).

The least squares fit generates an error; however, for the frequency range of interest, this error is small compared to the component uncertainty in f0 that is related directly to the uncertainty in the B&K microphone. A few of the magnitude and phase values of the B&K response with uncertainties from the LNE calibration are given in Table I. If the intent were to calibrate the Setra by direct comparison to the B&K, the results at low frequency would be disappointing: the B&K calibration uncertainty at 0.01 Hz is nearly 5% in magnitude and 5° in phase. However, we can use the comparison-derived response for the Setra and the least squares fit procedure to estimate the parameter, f0. One such determination gives a single value for f0. Several thousand determinations, each with its own perturbation of the B&K response according to its calibration uncertainty and each with its own least squares fit, form the basis for a Monte Carlo estimate of the uncertainty in f0.

TABLE I.

Calibration values and uncertainties for the BK4193 reference microphone. The umag and uphase are the k = 2 uncertainties in the B&K response.

Frequency Magnitude umag Phase uphase
[Hz] [mV/Pa] [%] [deg.] [deg.]
0.010  0.115  4.95  359.39  4.96 
0.020  0.336  2.68  303.42  2.19 
0.040  0.693  1.51  277.26  0.99 
0.079  1.299  0.93  248.56  0.48 
0.159  1.826  0.69  220.71  0.29 
0.316  2.065  0.58  201.82  0.23 
0.631  2.143  0.58  191.02  0.21 
1.260  2.155  0.58  185.47  0.21 
2.510  2.160  0.58  182.66  0.21 
5.010  2.160  0.58  181.16  0.21 
10.000  2.155  0.58  180.36  0.21 
19.950  2.150  0.58  179.97  0.21 
Frequency Magnitude umag Phase uphase
[Hz] [mV/Pa] [%] [deg.] [deg.]
0.010  0.115  4.95  359.39  4.96 
0.020  0.336  2.68  303.42  2.19 
0.040  0.693  1.51  277.26  0.99 
0.079  1.299  0.93  248.56  0.48 
0.159  1.826  0.69  220.71  0.29 
0.316  2.065  0.58  201.82  0.23 
0.631  2.143  0.58  191.02  0.21 
1.260  2.155  0.58  185.47  0.21 
2.510  2.160  0.58  182.66  0.21 
5.010  2.160  0.58  181.16  0.21 
10.000  2.155  0.58  180.36  0.21 
19.950  2.150  0.58  179.97  0.21 

Select a value for f0 and generate the Setra model response, H0_Setra, at each of the B&K calibration frequencies. In an arbitrary pressure field, P(f), the ideal Setra output would be P*H0_Setra. Generate the mean B&K response, H0_BK. The mean B&K output in that same pressure field is P*H0_BK. In an actual comparison calibration, each output would be measured and their spectral densities divided to cancel the applied pressure, P.

(This is overly simplistic for a real measurement. If the external pressure field is noise-like, then it is often better to construct the ratio by dividing the averaged cross-spectral density by the averaged auto-spectral density of the B&K. If the external pressure field is a high signal-to-noise ratio sine wave, it may be better to estimate the amplitude and phase of each output and construct the ratio from the ratio of amplitudes and the difference in phase. These practical considerations are not relevant in this discussion of ideal outputs.)

Cancelling the applied pressure gives the ratio of ideal responses. Multiplying this ratio by the mean B&K response would return the ideal Setra response. Instead, many samples of perturbed B&K response can be generated based on the mean values and uncertainties from the LNE calibration (see Figs. 2 and 3). Each sample of perturbed response (H1_BK) times the response ratio gives a sample of an equivalently perturbed Setra response (H1_Setra). By least squares fit to the phase (see the  Appendix), a sample of the characteristic-frequency (f0) estimate can be made for each H1_Setra. With many response samples, the mean and standard deviation of the many samples of f0 can be found.

FIG. 2.

Magnitude and phase response of the BK4193 infrasound measurement microphone with perturbations corresponding to the calibration uncertainties. The circles are the mean values, H0, and the solid lines are samples of the perturbed B&K response, H1. Ten realizations, each from 0.6 to 20 Hz, are shown here; 10 000 realizations were used in the Monte Carlo simulation. Over this frequency range, the k = 2 uncertainty (approximately the 95% confidence limit) is 0.6% in magnitude and 0.2° in phase (see Table I).

FIG. 2.

Magnitude and phase response of the BK4193 infrasound measurement microphone with perturbations corresponding to the calibration uncertainties. The circles are the mean values, H0, and the solid lines are samples of the perturbed B&K response, H1. Ten realizations, each from 0.6 to 20 Hz, are shown here; 10 000 realizations were used in the Monte Carlo simulation. Over this frequency range, the k = 2 uncertainty (approximately the 95% confidence limit) is 0.6% in magnitude and 0.2° in phase (see Table I).

Close modal
FIG. 3.

BK4193 response shown with the magnitude normalized by the mean response, |H1/H0|, and the phase as deviations from the mean phase, Δϕ = ϕ1 – ϕ0. The dashed lines are the k = 2 uncertainties from the LNE calibration: 0.6% in magnitude and 0.2° in phase. Ten realizations, each from 0.6 to 20 Hz, are shown. Each curve has 160 points. If the perturbations represent the 95% confidence limits, then, on average, 5% of the points would be beyond the dashed lines in each graph.

FIG. 3.

BK4193 response shown with the magnitude normalized by the mean response, |H1/H0|, and the phase as deviations from the mean phase, Δϕ = ϕ1 – ϕ0. The dashed lines are the k = 2 uncertainties from the LNE calibration: 0.6% in magnitude and 0.2° in phase. Ten realizations, each from 0.6 to 20 Hz, are shown. Each curve has 160 points. If the perturbations represent the 95% confidence limits, then, on average, 5% of the points would be beyond the dashed lines in each graph.

Close modal

Figure 4 shows the k = 2 uncertainties in f0 for 10 000 trials for each of several f0 values. The values of f0 and the associated uncertainty are in Table II.

FIG. 4.

Fractional uncertainty, u(f0)/f0 (plotted as percentage), in determination of the characteristic frequency, f0, based on the value of f0 and on the uncertainty in the BK4193 response.

FIG. 4.

Fractional uncertainty, u(f0)/f0 (plotted as percentage), in determination of the characteristic frequency, f0, based on the value of f0 and on the uncertainty in the BK4193 response.

Close modal
TABLE II.

Fractional uncertainty in the f0 component, u(f0)/f0, expressed in percent with k = 2 for determination of f0 in the Setra 278 model response as would be determined from comparison with the BK4193.

f0 u(f0)/f0
[Hz] [%]
10  0.137 
20  0.168 
30  0.209 
40  0.258 
50  0.309 
60  0.356 
70  0.412 
80  0.475 
90  0.520 
100  0.578 
f0 u(f0)/f0
[Hz] [%]
10  0.137 
20  0.168 
30  0.209 
40  0.258 
50  0.309 
60  0.356 
70  0.412 
80  0.475 
90  0.520 
100  0.578 

Having the uncertainty in f0 allows the resultant uncertainties in the magnitude and phase of the Setra response to be determined. The relevant equations are: (a) Eq. (7) for the magnitude uncertainty and (b) Eq. (11) for the phase uncertainty.

Figure 5 shows the combined uncertainty in the magnitude response for a Setra based on the uncertainty in DC response (0.08%, k = 2, from the manufacturer's datasheet) and on the uncertainty in f0 from Table II. The resulting combined uncertainty depends both on the component uncertainty in f0 and on the value of f0. Note that these uncertainties are the uncertainties associated solely with the Setra's DC uncertainty and the determination of f0. In a real measurement, uncertainties in the data-acquisition system, for example, would also enter the calculation.

FIG. 5.

The magnitude fractional uncertainty, u|S|, (converted to percentage) of the Setra 278 as a function of frequency. The four curves relate to four values of characteristic frequency, f0: 10 (largest uncertainty), 20, 50, and 100 Hz (smallest uncertainty; slight departure from 0.08% above 8 Hz). The combined magnitude uncertainty is a function of the uncertainties in DC response magnitude, S0, and in characteristic frequency, f0. Notice how small the combined magnitude uncertainty is as compared to the uncertainty in f0 (see Fig. 4). This is because the uncertainty sensitivity for uncertainty in f0 is well below 1.

FIG. 5.

The magnitude fractional uncertainty, u|S|, (converted to percentage) of the Setra 278 as a function of frequency. The four curves relate to four values of characteristic frequency, f0: 10 (largest uncertainty), 20, 50, and 100 Hz (smallest uncertainty; slight departure from 0.08% above 8 Hz). The combined magnitude uncertainty is a function of the uncertainties in DC response magnitude, S0, and in characteristic frequency, f0. Notice how small the combined magnitude uncertainty is as compared to the uncertainty in f0 (see Fig. 4). This is because the uncertainty sensitivity for uncertainty in f0 is well below 1.

Close modal

To illustrate the dependence on f0, uncertainty curves for four values of f0 are shown in Fig. 5.

The uncertainty in the Setra's phase response is shown in Fig. 6.

FIG. 6.

The phase uncertainty, uϕ, (converted from radians to degrees) of the Setra 278 as a function of frequency. The phase uncertainty is independent of the uncertainty in S0 [see Eq. (11)]. The uncertainty curves as functions of frequency are shown for four values of f0: 10 (largest uncertainty), 20, 50, and 100 (smallest uncertainty) Hz.

FIG. 6.

The phase uncertainty, uϕ, (converted from radians to degrees) of the Setra 278 as a function of frequency. The phase uncertainty is independent of the uncertainty in S0 [see Eq. (11)]. The uncertainty curves as functions of frequency are shown for four values of f0: 10 (largest uncertainty), 20, 50, and 100 (smallest uncertainty) Hz.

Close modal

While the process of bridging the gap in pressure response from near-zero frequency to a frequency high enough to overlap with an AC infrasound sensor is general, this paper considers a specific case of bridging sensor: the Setra 278 barometric-pressure sensor with demonstrably simple low-frequency response. Another sensor could be used as the bridging sensor as long as (1) that sensor has response at zero frequency, (2) a model exists for the response over the gap in frequency from zero to the lower calibrated limit of the AC infrasound sensor, and (3) the parameters of the bridging-sensor model can be determined by comparison with the AC sensor. The second point can be problematic as manufacturers are often reluctant to disclose details of the electronics internal to their sensors. Another issue deferred for future study is temperature dependence of the Setra 278's characteristic frequency, f0.

The first step in establishing the Setra 278 as a reference for comparison calibration is to find the characteristic frequency and its uncertainty. This step requires comparison to an infrasound sensor having its own traceable calibration. A calibrated BK4193 was used for the analysis in this paper. The next step is to find the uncertainty in magnitude and uncertainty in phase for the Setra based on the value and uncertainty of f0. The Setra can then be used for comparison calibrations of other infrasound sensors without repeating the steps that require the calibrated infrasound sensor.

From a comparison of Figs. 5 and 6 to Table I, notice that the uncertainty in both magnitude and phase for the Setra response is considerably lower than the uncertainty associated with the response of the B&K reference even though the B&K is used to determine the f0 parameter of the Setra model response. The low sensitivity of the Setra response to uncertainty in the parameter, f0, and the low sensitivity of f0 to uncertainty in the infrasound comparison sensor (BK4193 here) are the key elements.

The principal features of this process are that the resulting uncertainty in the Setra response from DC to 10 Hz is low, a traceable estimate for S0 is available from the manufacturer, and f0 can be determined in a traceable manner. For comparison calibration, in which reference-sensor self-noise is not an issue, the Setra, or another DC sensor like the Setra, may prove to be a viable option for the DC-to-10 Hz frequency range with the potential for better than several tenths of a percent (k = 2) uncertainty in magnitude and better than a few tenths of a degree (k = 2) uncertainty in phase if f0 is greater than 20 Hz and the frequency of interest is below 10 Hz.

See the supplementary material for an unpublished report by one of the authors (C.L.T.) in the file SuppPub1.pdf.

This work was supported by the National Institute of Standards and Technology under Award No. 60NANB23D165.

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material. Please contact C.L.T. ([email protected]) for data requests.

The magnitude of the response of the Setra in the limit as frequency goes toward zero is given, along with its uncertainty, by the manufacturer through traceable calibration. With that value, knowledge of the phase of the response with frequency is sufficient to define the complex frequency response. If we assume that the phase response can be estimated at a number of frequencies (typically, by comparison calibration) and we assume the validity of the two-pole low-pass character of the response, an estimate for the only undetermined parameter (f0) can be made through least squares fitting of the low-pass phase response against the measured phase. As discussed in the text, the uncertainty of the estimate for f0 can have considerably smaller bounds than the uncertainty of any single point in the comparison-reference transducer. A critical step, then, is the least squares fit.

The model relationship for the response phase is
(A1)
Also, required is the derivative of the phase with respect to f0,
(A2)
Let ϕm(f) be the phase of the measured data. Then the goal is to minimize the difference, Z, between the measured phase and the model phase,
(A3)
The first-order approximation—which is used to guide the iterative search for best fit—to this difference is
(A4)
For N values of frequency in the measurement of phase, there will be N values of Z and of the partial derivative; consequently, Eq. (A4) is a set of N equations in one unknown (Δf0). The solution, by pseudo-inverse (“pinv”) of the N × 1 partial-derivative matrix, gives the least squares estimate for Δf0,
(A5)

The iteration to reduce Δf0 (and, of course, Z) toward zero proceeds as follows: (1) make an initial guess for f0, the value of which is not critical; (2) construct the Nx1 matrix of partial derivatives using Eq. (A2); (3) construct the difference matrix, Z; (4) find the correction, Δf0, using Eq. (A5); (5) update the value for f0, by subtracting the correction, Δf0; (6) find the mean square of the difference, Z; (7) test that mean square—if close enough to zero, stop; otherwise, repeat steps (2)–(7).

The pseudo-inverse of an N × M matrix, A, is
(A6)
where inv is the ordinary matrix inverse and the superscript, T, indicates the transpose. For an N × 1 matrix this simplifies to
(A7)
where the denominator on the right side is a scalar: the sum-square of the elements of A.
The mean square error of the least squares fit is
(A8)

Once the process has converged to a stable value for f0, the magnitude of the model response should be calculated and compared to the measured magnitudes. If the process has converged properly (and the two-pole model is appropriate), the predicted magnitude response should be close to the measured magnitudes.

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Supplementary Material