Field calibration with seismic stimulus (Gabrielson-Charbit)

Station operators from IMS and other seismological networks now routinely conduct calibrations by comparison with a co-located reference. This method was first proposed for seismometers by Pavlis & Vernon (1994) and for infrasound sensors by Gabrielson (2011). The Gabrielson approach has been widely implemented for IMS infrasound stations in recent years.

For field calibration of seismometers with seismic stimuli, an adapted version of the Gabrielson approach with modifications from Charbit et al. (2015) and Green et al. (2021) is employed as this is supported and already implemented by the CTBTO/PTS for IMS infrasound stations.

The calibration principle for seismometers starts with the assumption that the convolutional model for the recorded seismograms is valid (see section “calibration methods – calibration with seismic stimuli”; Pavlis & Vernon, 1994). The transfer or gain function Z(ω) is related to the spectral ratio Si(ω) of the recorded signals and independent of the ground motion. This is similar to Gabrielson’s approach, where the knowledge of the ratio of the responses Z=ISUT/IREF, the so-called gain ratio, is sufficient to find the unknown response \(I_{\mathrm {SUT}}\):

\(\hat I_{\mathrm {SUT}}(\omega) = \hat Z(\omega)\cdot\hat I_{\mathrm {Ref}}(omega)\)

The determination of the estimated transfer function \(\hat Z(\omega)\)  between the sensors is done for example by calculation of spectral ratios (e.g. Pavlis & Vernon 1994). Gabrielson suggests to use averaged auto spectral (\(\hat G_\mathrm{SutSut}\)) and cross-spectral (\(\hat G_\mathrm{SutRef}\)) densities to reduce bias and scatter due to noisy signals. By this, incoherent signals are averaged towards zero.

\(I_\mathrm{SUT} = \frac{\hat G_\mathrm{SutSut}}{\hat G_\mathrm{SutRef}^*} I_\mathrm{Ref}\)

where the * denotes the complex conjugate. For this approach the following conditions apply:

1. The reference sensor/transfer standard has a known and traceable frequency response function (this ensures traceability for the field sensor calibration)

2. The sensor under test and the reference sensor measure the same, coherent signal.

3. Effects of incoherent signals are assumed to be negligible.

Assumptions ii and iii are crucial for this technique. Since both sensors also detect incoherent signal components, it is necessary to take measures that ensure the response is determined for signals that are observed by both seismometers with high coherence and similarity.

Gabrielson (2011) suggests using the magnitude squared coherence (MSC) γ² as a similarity measure:

\(\gamma^2 = \frac{\hat G_\mathrm{SUTRef} \hat G_\mathrm{SUTRef}^*} {\hat G_\mathrm{SUTSUT} \hat G_\mathrm{RefRef}}\)

By using only frequencies and time intervals where γ² is greater than a specific threshold, intervals with low coherence are excluded from the analysis. In practice, a threshold of 0.98 has proven to be a good choice (Charbit et al., 2015), as it satisfies assumption ii. However, good results can be obtained if the threshold is greater than 0.8 depending on the type and distribution of incoherent signals.

As a second similarity measure the usage of the Pearson cross-correlation coefficients between the recorded signals of sensor under test and reference sensor is introduced for seismometers. This additional measure is necessary as satisfactory outcomes cannot be obtained solely with MSC. A small MSC threshold value (e.g. <0.8) causes considerable scatter in the gain ratio. However, a MSC threshold value of 0.98 results in insufficient coherent data, especially at low frequencies (<0.7 Hz). The cross-correlation rxy between two time series x and y with a time-lag of τ, is defined as

\(r_\mathrm {xy}(\tau) = \sum_{t=t_0} ^{t_1} f_{\mathrm x ,t} \cdot f_{\mathrm y ,t}\)

of which the maximum value is taken. A typical threshold for this measure is 0.8. An additional benefit of calculating the cross-correlation is the ability to obtain the time lag between the two time series. This lag is helpful in checking for time synchronisation errors.

If the reference and test sensor are installed far apart or more than one sensor is to be calibrated with a single reference, another similarity measure is necessary. The suggested measure is based on the multichannel measure of coherence ρmax, which was first introduced by Green et al. (2021) for infrasound station calibration. Although this multichannel measure is effective, it comes with high computational costs due to the typically higher sampling rate and possibly larger number of seismometers in a regional array compared to infrasound stations. Following Green et al. (2012), first Rmax, the maximum of the cross-correlation between each sensor pair ij for M sensors is calculated:

\(R^\mathrm{max} = \left[ \begin{array}{ccc} \mathrm{max}(r_{11}) & \cdots & \mathrm{max}(r_{1M}) \\ \vdots & \ddots & \vdots \\ \mathrm{max}(r_{M1}) & \cdots & \mathrm{max}(r_{MM}) \\ \end{array} \right] ,\)

with rij being the cross-correlation between sensors i and j. The multichannel measure of coherence is given by

\(\rho_\mathrm{max} = \frac {\sum_{i=1}^{M}\sum_{j=1}^{M} R_{ij}^\mathrm{max}} {M \sum_{i=1}^{M} R_{ii}^\mathrm{max}} .\)

The numerator is the sum of all elements of  \(R^\mathrm{max}\), the denominator is given by the trace of \(R^\mathrm{max}\) multiplied with the number of sensors used.

These similarity measures help to identify signals that are observed by both the reference and test seismometers. A further constraint on the method is the stationarity property, which is needed for the spectral approach. Charbit et al. (2015) extended Gabrielson's (2011) approach by dividing the signal into six passbands and using varying data segment lengths (Table xx) to calculate the power spectral density in each passband. This approach ensures the time series are approximately stationary and allows the use of spectral averages over shorter time windows to estimate response calculations at higher frequencies. This increases the likelihood of identifying windows with low noise and allows more spectral estimates to be averaged to produce the response estimate.

Table xx:

* last upper cutoff frequency depends on the sampling rate of the seismometer. For the short-period seismometers (e.g. GS13) the sampling rate is 40 samples/s, therefore, to fulfil the Nyquist criterion, the upper cutoff frequency must be below 20 Hz.

Implementation of the method:

1. The raw data time series of both sensors are filtered within the passbands (Butterworth bandpass filter)

2. The filtered data is subsequently divided into segments of different lengths, depending on the passband

3. Within each segment, averaged cross- and auto-spectral densities are computed using Welch’s method (Welch, 1967) to sub-divide each segment into 9 windows with an overlap of 50% (Hanning window)

4. Within each segment, the similarity measures are computed 

5. If the similarity measures are greater than pre-defined thresholds, then the gain ratio or complex transfer function between the sensor under test relative to the reference seismometer is determined by
   \(\hat Z = \frac{{\hat G}_{\mathrm SUTSUT}}{{\hat G}^*_{\mathrm SUTRef}}\)

6. This process is repeated for all segments within each passband

7. From all determined gain ratios, a single gain estimate (ω) for each frequency is computed by a weighted mean
   \(\bar g_{\gamma^2} =\frac{\sum_{n=1}^N w_n \hat Z_n } {\sum_{n=1}^N w_n} =I_{\mathrm cal}\)
   \(N\) is the number of gain measurements \(G_n\) .
   The weight \(w\) is given as
   \(w = \left( \frac{1}{2\cdot(2P+1)} \frac{S_{\mathrm SUTSUT}}{S_{\mathrm RefRef}} \left( \frac{1-\gamma^2}{(\gamma^2)^2} \right) \right)^{-1}\)
   where 2P + 1 = 9, is the number of periodograms/windows averaged to provide the spectral density estimates. (Note: high MSC thresholds have only little effects; the usage of the weight is of greater importance when smaller thresholds are used)

8. With the known frequency response of the reference the response of the sensor under test can be determined
   \(I_{\mathrm SUT} = I_{\mathrm Ref}\cdot I_{\mathrm cal} \)

9. As multiple gain measurements are made at each frequency, uncertainties can be assigned to the amplitude and phase estimated using the weighted standard deviation of the gain in amplitude and phase: 
   \(\sigma_{\mathrm A} = \sqrt \frac{w_n(|G_n|-|\bar g_{\gamma^2}|)^2}{\sum_{n=1}^N w_n}\)
   \(\sigma_{\mathrm \phi} = \sqrt \frac{w_n(arg(G_n)-arg(\bar g_{\gamma^2}))^2}{\sum_{n=1}^N w_n}\)
   where \(\sigma_{\mathrm A}\) and \(\sigma_{\mathrm \phi}\) are the amplitude and phase standard deviations, respectively.

10. The usage of time series of one day length are recommended. This is advantageous in avoiding large data files. A single gain value and uncertainties can be obtained through averaging over all days within the considered time range.

One requirement for the chosen on-site calibration approach is the record of sufficient coherent excitation signals above the self-noise levels of both the reference and test seismometers within the relevant frequency range. Schwardt et al. (2022) identified suitable excitation sources for on-site calibrations based on a literature review of a large variety of both anthropogenic and natural sources of seismic signals regarding their frequency bandwidth, signal properties, and applicability in terms of cost–benefit. Besides some previously applied natural sources (ground noise, microseisms, Earth’s tides), anthropogenic (e.g. cultural noise) and man-made controlled sources (e.g. drop weights, hammer blows, vibration sources) fulfil the pre-defined conditions for excitation signals to a large extent. The usage of controlled sources is advantageous as signals may be generated with great repeatability, high energy, and within a defined frequency range, the latter depending on the source design. Due to them being portable, these sources may be applied for identifying any misalignment between the sensors or directional influences.

Based on Schwardt et al. (2022) many sources appear to be suitable and earthquakes stand out as a result of their signal properties, but there are many factors that influence their application:

1. Frequency range:

The bandwidths for seismic sensors specify the frequency ranges in which the response is flat to ground velocity. For broadband sensors (e.g. STS2.5), the flat-to-velocity portion of the bandwidth generally ranges between 0.01 Hz to > 25 Hz, whereas for short-period seismometers (e.g. GS13) it ranges from 1 Hz to 100 Hz. Within the project, a frequency range between 0.01 and 20 Hz was considered. Therefore, sources that excite frequencies within this range are of interest.

2. Signal length:

The signal length is determined by the lowest frequencies to be calibrated. The lower the frequency, the longer the record length. According to the applies calibration approach, the longest signal needs to be at least 10 times as long as the longest period (smallest frequency) to be calibrated.

3. Signal strength:

the recorded signals must be well above the instrumental self-noise (e.g. electrical) and background seismic noise levels, the latter being station depend.

4. Applicability:

This property includes aspect such as cost-benefit when controlled sources are concerned and signal repeatability for natural sources.

Relying on only one type of excitation signal is impractical, because of the repetition rates of earthquakes of certain magnitudes or the limited frequency ranges of individual sources (e.g. microseismicity). We therefore suggest the usage of all data recorded by the seismometers and the exclusion of signals which do not meet the predefined similarity criteria during processing. Thereby, all frequencies within the relevant range are covered to a certain extend and the response is determined for all kinds of relevant ground motions.

Figure: Observed frequency ranges for different sources of seismic waves. Dashed-bordered boxes illustrate anthropogenic sources, solid-bordered boxes illustrate natural sources. More saturated colours indicate commonly observed and dominant frequency ranges. The frequency ranges to be calibrated (0.01–20 Hz for seismic waves) are highlighted in grey. Note that only the most important and not all sources are included in the figure, and sources, which are well outside the frequency range under consideration, have not been included for reasons of clarity. The figure is adapted from Schwardt et al. (2022).

Transportation of reference device/Transfer standard:

The transfer standard/reference sensor must be transported to/from the station by car/plane without any risk of damage. The usage of a robust sensor with specific packaging (usually provided by the manufacturer) in addition to mechanical or electrical locking of the sensitive parts such as the mass are recommended in order to limit the risks of damage.

Whether and to what extent the repeated transport of the sensor between the station and a laboratory influences the calibration is still being investigated.

Instruments:

Both the test and reference sensor should have a similar bandwidth (flat to velocity) and based on the same operating principle. It is possible to compare different sensor types with each other.

Co-Location:

Co-location is of great importance to ensure that the reference and test sensor record the same coherent signal. Co-location in this context means that the sensors should be as closely placed as possible to each other, but may also be installed at distances of 5-10 m from each other, e.g. in neighbouring vaults. There are many factors that have an impact on the co-location. On the one hand, there is the available space (in a vault), on the other hand, there are site characteristics to consider.

There is a directionality in seismic signals because the waves propagate in 3D space. If the sensors are too far apart, one sensor receives the signal before the other, resulting in a time shift, which affects the coherence. If necessary, the time lag can be determined by cross-correlation and the signals can be shifted to be synchronized. However, if the distance between the sensors is larger (n*100m), the subsurface may affect the signal shape and thereby the coherence/similarity. Therefore, a direct co-location is preferable whenever possible.

However, due to certain signals and their properties, it is possible to calibrate reference and test sensors that are further away from each other (max. 1500 m) (see sections “Calibration with Seismic Stimuli” and “Possibility of station-wide calibration”).

Installation:
 When installing the reference sensor, there are a few things that need to be considered. These include the orientation of the sensor, the levelling, and the alignment of the reference sensor in relation to the test sensor. Orientation and levelling also apply to the installation of the test sensors.

Levelling:

Must have a means to control levelling such as an integrated physical or digital bubble level. This mean must offer good visibility during on-site implementation to check horizontal alignment. The sensors must have at least one level indicator to minimize tilting effects as much as possible, for example locking feet.

Alignment & Orientation:

Sensors should have some easily identifiable scribe marks and accessories to identify the orientation of the horizontal axis (E/W) in order to minimize the effect of the misalignment between two or more sensors. A mean for alignment with true north such as an orientation/alignment rod should be used. The true north should be determined by a mean (compass) and marks on the ground/walls are preferable for a later installation. For the IMS station seismometers an orientation error of less than 3° must be guaranteed. A possible misalignment between reference and test/field sensor can be checked later on the basis of the measured data. A detailed description about the determination of sensor alignment is given by Holcomb (2002).

Digitizer:

If possible, both the field/test and the reference sensor should be connected to the same digitizer to avoid any influences of the digitizer on the calibration results and effects of non-synchronization. If two separate digitizer must be used care needs to be taken regarding the time synchronization. Minimal differences in the time synchronization already cause errors in the phase. The PTS requires an absolute timing accuracy of each sensor to be less than 10ms deviation from Coordinated Universal Time (UTC) and a relative timing accuracy of less than 1ms in signal timing difference between array elements.

Additionally, the digitizers needs to be traceable calibrated. When using the same digitizer for both the reference and field/test seismometer its transfer function is eliminated from the analysis.

Pre-Amplifier:

Some sensors require a pre-amplifier. As with the digitizer the pre-amplifier should be traceable calibrated beforehand to ensure a traceable seismometer calibration in the end.

Timing:

If the sensors or sensor systems are not exactly synchronized in time, the analysis results in incorrect phase determination, especially in the higher frequency range. A timing error appears as a linear deviation of the phase with increasing frequency (linear frequency axis). A first check for such a behaviour is the calculation of the time lag between both time series (e.g. by using cross-correlation) within each frequency band and for the unfiltered time series. Under the assumption that both sensor record the same coherent signal and are closely located, there should be no time difference between the signals.

The shift of the phase caused by timing errors is given by

with the time difference tdelay (lag) between the signals in seconds and the frequency f in Hz. Knowing the time delay between the signals and the resulting phase shift, the phase can be corrected:

A small timing error of 0.1 samples (0.001 s with a sampling frequency of 100 Hz) results in a phase shift of 0.0036° at a frequency of 0.01 Hz. The same timing error causes phase shifts of 0.036, 0.36, and 3.6° at frequencies of 0.1, 1, and 10 Hz, respectively. These values all lie within the 5° tolerance. Likewise, a timing error of 0.001 s satisfies the relative timing accuracy between array elements. However, test have shown that at least a time delay of 0.005 s (0.5 samples) is to be expected.This indicates that the time bases may be synchronised, but the exact sampling times are not. In this case, both channels would have a random time between 0 and 1 sample between the defined start time and the sampling time of the first sample, depending on the exact time at which the A/D converter started sampling. On average, this would be half a sample difference between two tracks from different digitisers. Note that time delays may also be caused by the distance between co-located seismometers and seismic waves reaching one sensor before the other.

Figure: Example for the phase response of the test sensor with a timing error. The linear deviation of the calculated phase values (blue dots/line) from the nominal values (red) is shown in a) with a linear frequency axis. The same is shown in b) for a logarithmic frequency axis including some corrected phase values (black stars).

Power supply:

Should be guaranteed (e.g. be means of emergency power), preventing any failures and loss of data.

Sampling rates:

Ideally, both sensors should sample the signal at the same sampling rate. Different sampling rates between field/test and reference sensor require an adjustment to an equal sampling rate for analysis. This could be done by re-sampling the data. However, re-sampling the data is not readily achievable and a couple of aspects need to be taken into account. When down sampling is applied, the amount of data is reduced, which is, to some extent, advantageous. Yet, this is also accompanied by a decrease in resolution and a loss of higher frequencies that might be of interest. the highest frequency to be considered here is based on the Nyquist criterion.

Re-sampling alters the data, e.g. through integrated anti-alias filters. This may effect the result of the calibration. Further, it must be ensured that the samples are at exactly the same time.

Excitation signals:

As stated above; various types of signals can be used for on-site calibration. The usage of the whole record is suggested as all frequencies within the relevant range are covered to a certain extend and the response is determined for all kinds of relevant and station-typical ground motions.

Similarity measures:

Several similarity measures are used to select time periods/signals from the entire record that meet the selected similarity criteria. This ensures that coherent signals are used in the analysis. In addition to the magnitude squared coherency (MSC), the use of cross-correlation is also recommended for seismic data. The thresholds for both measures are chosen based on experiences but should be adapted to the respective stations through testing. For the MSC a threshold value of 0.98 has proved suitable, for the cross-correlation values greater than 0.8 are recommendable.

An additional similarity measure based on the cross-correlation is used in the case of station wide analysis or analysis of sensors with a greater separation. This measure is related to the station wide similarity measure proposed by Green et al. (2021) for infrasound data.

Figure: Cross-correlation coefficients for the comparison of two co-located 3-channel broadband seismometers for time series of a days length for a total of 260 days. The cross-correlation coefficients are calculated for each channel/axis of the seismometer, red dots show the values for the vertical (Z) axis, blue and black for both the horizontal axes, respectively. The vertical grey dashed line marks the end of 2022, the horizontal grey dashed line marks a coefficient of 0.8, which was chosen as threshold value. Especially for the horizontal axis the values drop below the threshold at the end of November and begin to get larger again around April. This demonstrates, how important it is to have either continuous on-site calibration capabilities or chose a suitable time beforehand. Note that here the values are shown for an unfiltered time series of a whole day, whereas in the method these were calculated within each window for the filtered data.

Figure: The coherency for frequencies between 0.01 and 20 Hz is shown for two co-located 3-channel broadband seismometers for time series of a days length for a total of 260 days. It is apparent, that the time series are highly coherent for frequencies greater than 0.7 Hz for all three axes. Similar to the cross-correlation coefficients, the coherency changes to lower values for frequencies below 0.7 Hz between November and December 2022, especially for the horizontal axes. This demonstrates that not all times of the year at a particular station are suitable for the on-site calibration approach. Note that here the values are shown for an unfiltered time series of a whole day, whereas in the method these were calculated within each window for the filtered data.

Figure: The percentage of used segments per considered frequency is shown for each month. As the values of the applied similarity measures drop below the threshold (see other figures), less data is used. For frequencies between 0.06 and 5 Hz between approximately 45 and 65% of available data segments could be used for the analysis, whereas this drops below 20% for the other months.

Filter Banks:

The analysis is undertaking across eight passbands, which are adopted from Charbit et al. (2015) and Green et al. (2021) and extended to cover the higher frequency range of seismometers. This approach ensures that the time series are stationary and increases the probability of identifying times of high coherence. Based on this methodology, the passbands can also be adapted, e.g. to the frequency ranges of typically registered waveforms or frequency filters used during the analysis for earthquake/seismic signal interpretation.

Duration:

The duration of the measurement depends on the considered frequencies and the recorded signals. The lower the frequencies to be calibrated, the longer the duration of the experiment. Additionally, the duration depends on the signals recorded at the station: there might be periods, when there is less coherent signal than at other times, as some signals might occur seasonal. A minimum duration of at least 14 days is recommended. Furthermore, longer periods of time provide a larger sample size for averaging (see previous figure), more stable results (see figure below), and allow for the investigation of seasonal fluctuations in the station's behaviour, such as temperature differences or variations in dominant signals.

Figure: Scattering of calculated ratios for a time series of one day (left) and 15 days (right) for both amplitude (top) and phase (bottom). The nominal value for the sensor under test is shown as red curves, the weighted mean as orange stars/curves. Grey shaded areas show the distribution of all values that are included in the averaging. The darker the shaded area, the more values lie within these limits. 

Influence between co-located instruments?

Does one sensor affect the other if they are too close together? (electronic noise? tilt? ...)

Further investigation is needed for this topic.

Possibility of station-wide calibration?

A station often consists of an array of seismic sensors. So far, the calibration of a single sensor with a co-located reference applying the Gabrielson method is discussed. The question is whether more than one element/sensor of the array can be calibrated by one reference or, in other words, how far can sensors be from the reference instrument and how many reference sensors are needed to calibrate all sensors of an array without the need of too often re-localizing the reference within the array?

For the first experiment, two calibrated vertical short-period seismometers were installed at a distance of approximately 1400 m. Using one of these sensors as a reference and the other as the sensor under test (as well as the other way around), the amplitude and frequency response obtained by the Gabrielson method (blue lines) is compared with the laboratory values (black asterisks).

For frequencies below 1 Hz, the amplitude response values of the test sensor fit well with the laboratory values. However, there are deviations larger than 5° observed for the phase response. Note that when the reference – test sensor pair are exchanged (right column), the phase deviations are mirrored. This indicates an influence of the travel time of the signal between the sensors.

In a subsequent step, one calibrated sensor is used as reference for all other short period seismometers of the array. Following the Gabrielson method, the complex gain ratios and the corresponding amplitude and phase responses were determined and averaged for days that showed high cross-array coherency values. The preliminary results show that for low frequencies (f ≤ 1Hz) promising results can be obtained, especially regarding the amplitude response. The phase response of one sensor lies within the expected range, whereas the other two sensors show large deviations from the expected values. This might be caused by the distance between the sensors. As the signals travel across the array, they are recorded by the sensors with a time shift relative to their distance, resulting in a phase shift in the response determination. This needs further investigation.

The coherency is a good indicator of the distance between the field and reference sensors, as it decreases with distance. In all probability, there is no need for an individual reference for each seismometer, but an array station can be calibrated with a significant smaller number of reference sensors. Note that this number is both site as well as signal dependent and requires further investigation.

Environmental Influences:

What is the influence of temperature, pressure, humidity, etc. on the sensors? It is recommended that the reference sensor has been calibrated under the same or similar conditions as the field/test sensor or that its behaviour under different environmental influences is known. This is of particular interest when using different sensors as reference and test sensor as they might behave differently under changing conditions.

If the sensor's response and the reference sensor's response change similarly due to environmental factors such as temperature, pressure or humidity fluctuations, an in-situ technique will most likely fail to detect these changes.

The measurement uncertainty of the field calibration inherits component from the concatenated calibration processes from the laboratory to the field. In addition to the estimated measurement uncertainty in the seismometer sensitivity itself, converting the analogue signal to digital via the digitizer introduces extra uncertainty into the measurement chain. Environmental factors, including temperature or pressure, must be taken into account during field calibration. Other sources of uncertainty include the levelling and orientation of the seismometers and their alignment to each other. Only the uncertainties of the instruments themselves will be discussed in more detail below. The other mentioned influences, while equally important, are very much situation-specific, and difficult to quantify in general, so are therefore not considered at this stage.

In order to obtain the response ISUT of the field seismometer the relative transfer function Ical is multiplied with the known response function IREF of the reference:

\(I_{\mathrm SUT} = I_{\mathrm Ref}\cdot I_{\mathrm cal}\)

The frequency response\(I_{\mathrm cal}\) of the sensor under test relative to the reference seismometer is obtained by the modified Gabrielson method as described above (see “field calibration – implementation of the method”). The response \(I_{\mathrm Ref}\) of the reference is known from laboratory calibration including uncertainties for both the amplitude (\(u_{a_0}\)) and phase (\(u_{\varphi_0}\)). Note that so far the complex frequency response was considered, which can also be expressed in terms of amplitude and phase:
\(I_j = A_j \cdot e^{i \phi_j}\)

The amplitude and phase of the test sensor's response are estimated by the following equations:
\(A_{\mathrm SUT} = A_{\mathrm Ref}\cdot A_{\mathrm cal}\)
\(\phi_{\mathrm SUT} = \phi_{\mathrm Ref} + \phi_{\mathrm cal}\)

With the uncertainties provided for the reference instrument in the calibration certificate, the uncertainties for the test sensor can be determined under the assumption that the responses are considered to be Gaussian distributions:

\(u_{A_{\mathrm SUT} } = \sqrt{A_{\mathrm SUT}^2 \left(\left(\frac{\sigma_{A_{\mathrm cal}}}{A_{\mathrm cal}}\right)^2 + \left(\frac{u_{A_{\mathrm Ref}}}{A_{\mathrm Ref}}\right)^2 \right)}\)

\(u_{\phi_{\mathrm SUT} } = \sqrt{\sigma_{\phi_{\mathrm cal}}^2 + u_{\phi_{\mathrm Ref}}^2 }\)

The uncertainties \(\sigma_{\mathrm cal}\) for both phase \(\phi_{\mathrm cal}\) and amplitude\(A_{\mathrm cal}\) of the relative response are given by the standard deviation obtained by equations (x & y).

References (field calibration)

Pavlis, G. L., & Vernon, F. L. (1994). Calibration of seismometers using ground noise.Bulletin of the seismological society of America, 84(4), 1243-1255.

Gabrielson, T. B. (2011). In situ calibration of atmospheric-infrasound sensors including the effects of wind-noise-reduction pipe systems. The Journal of the Acoustical Society of America, 130(3), 1154-1163.

Charbit, M., Doury, B. & Marty, J., 2015. Evaluation of infrasound insitu calibration method on a 3-month measurement campaign, in 2015 Infrasound TechnologyWorkshop of the Provisional Technical Secretariat of the Comprehensive Nuclear-Test-Ban Treaty Organisation, Vienna International Centre, Vienna.

Green, D. N., Nippress, A., Bowers, D., & Selby, N. D. (2021). Identifying suitable time periods for infrasound measurement system response estimation using across-array coherence. Geophysical Journal International, 226(2), 1159-1173.

Welch, P. (1967). The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Transactions on audio and electroacoustics, 15(2), 70-73.

Schwardt, M., Pilger, C., Gaebler, P., Hupe, P., & Ceranna, L. (2022). Natural and anthropogenic sources of seismic, hydroacoustic, and infrasonic waves: Waveforms and spectral characteristics (and their applicability for sensor calibration). Surveys in Geophysics, 43(5), 1265-1361.

Holcomb, L. G. (2002). Experiments in seismometer azimuth  determination by comparing the sensor signal outputs with the signal  output of an oriented sensor. US Department of the Interior, US Geological Survey.