National Physical Laboratory

Uncertainties in Free-field Hydrophone Calibration

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To be truly meaningful, the result of a calibration must be accompanied by its associated uncertainty. In general, uncertainties are distinguished by how the values are estimated:

Type A: evaluated by statistical means (sometimes called 'random uncertainties')
Type B: evaluated by other means (sometimes called 'systematic uncertainties')

Type A uncertainty

This may be obtained from a statistical analysis of the repeatability of the calibrations.

Ideally, the repeated measurements should be truly independent repeats, with the hydrophones removed from the water and remounted before the calibration is repeated.

Where it is not feasible to undertake such independent repeats, and where historical data exists for the repeatability of the measurements with the devices in question, values for the typical repeatability for calibration of a hydrophone may be used.

Type B uncertainties

Type B components of uncertainty are those that are not assessed by statistical means, in other words those errors that remain constant when the measurement is repeated. For example, any systematic bias in a measurement may be regarded as a Type B contribution. Similarly, the uncertainty in the calibration of a calibrated instrument provides a Type B contribution.

The sources of these components must be identified by assessing all of the influences that may introduce errors into the measurement. These will be different for each measurement system and must be assessed individually. The value of each component must be estimated along with an associated probability distribution.

Common sources of Type B uncertainty

The following is a list of common sources of systematic uncertainty in the calibration of underwater acoustic hydrophones. The list is not exhaustive, but may be used as a guide when assessing uncertainties for a specific implementation of a calibration method. Depending on the calibration method chosen and its implementation, some (though possibly not all) of these sources will need assessing. For example, the errors from measuring instruments may be minimised by the use of the same measuring channel (amplifier, filter, voltmeter, etc) for all signals and measuring only amplitude ratios. However, since this may not be the case in all implementations, components for these sources of error have been included in the list.

Uncertainties specific to free-field reciprocity calibrations:

  • Inaccuracy of any assumptions about the acoustic field, eg that the field is a spherical-wave field (this may be checked by varying the separation distance between transducers and checking that the product of transfer impedance and distance is invariant)
  • non-reciprocal behaviour by transducers (can be evaluated by checking the equivalence of the ZPT and ZTP transfer impedances)
  • errors in the measurement of the separation distance
  • errors in the values for acoustic frequency (required to calculate the reciprocity parameter)
  • errors in the values for water density (required to calculate the reciprocity parameter)

Uncertainties specific to comparison calibrations:

  • Errors in the calibration of the reference hydrophone (a major source of error in comparison calibrations)
  • short term stability of any auxiliary transducers used for comparison calibrations (eg stability of the output of a transducer used as a projector in a comparison calibration)
  • stability of the reference hydrophone, (variation in sensitivity of reference device since previous absolute calibration). This can be assessed by repeating the measurements using a different reference hydrophone - it is very unlikely that both reference devices would be unstable to the same degree
  • differences in environmental conditions for the comparison calibration compared with those that existed during the absolute calibration of the reference hydrophone causing a change in sensitivity for the reference hydrophone (eg temperature, depth, mounting/rigging)

Uncertainties common to both methods:

  • Lack of steady-state conditions, especially where bursts of single-frequency sound waves are used - the resonance frequency and Q-factors of the transducers and the echo-free time of the tank will influence this contribution which will tend to increase at lower frequencies
  • interference from acoustic reflections, leading to a lack of free-field conditions
  • lack of acoustic far-field conditions
  • the spatial averaging effects of the hydrophones under calibration due to their finite size and the lack of perfect plane-wave conditions
  • misalignment, particularly at high frequencies where the hydrophone response may be far from omnidirectional
  • acoustic scattering from the hydrophone mount (or vibrations picked up and conducted by the mount)
  • errors in the measurement of the receive voltage (including the accuracy of the measuring instrumentation - voltmeters, digitisers, etc)
  • accuracy with which the gains of any amplifiers, filters, and digitisers are known
  • errors in the measurement of the drive current or voltage
  • errors due to the lack of linearity in the measurement system (the use of a calibrated attenuator to equalise the measured signals may significantly reduce this contribution)
  • accuracy of any electrical signal attenuators used
  • electrical noise including RF pick-up
  • accuracy of any electrical loading corrections made to account for loading by extension cables and preamplifiers
  • bubbles or air clinging to transducers - this should be minimised by adequate wetting and soaking of transducers
  • environmental conditions such as water temperature and depth of immersion - corrections need not be included for these if the calibration results specify the conditions and state that the calibration is only valid for the conditions stated

Overall uncertainties

The overall uncertainty should be obtained from the Type A component and the individual Type B components by combining the values in quadrature. All components must be expressed as standard uncertainties before being combined to obtain the overall uncertainty.

When stating the combined overall uncertainty, it is common to express the uncertainty as an expanded uncertainty. In this case the level of confidence and the coverage factor must also be stated.

When combining uncertainties, care should be taken when component values are expressed in decibels. Before combination, the values should be expressed in linear form (eg in percent) and not in decibels (dB). The final value of expanded uncertainty may then be expressed either in percent or converted to decibels as required.

Note: It should be realised that the use of decibels to express uncertainties may lead to asymmetric distributions (eg +1.5 dB is equivalent to +19%, but -1.5 dB is equivalent to -16%).

With care and when using hydrophones within their main operating range, it is possible to achieve an overall expanded uncertainty for a coverage factor of k=2 (confidence level of 95%) which is significantly better than ±10% (approximately ±0.9 dB).

Further guidance

Further guidance for evaluating and expressing the uncertainty in measurements can be found in the following documents:

  1. A Beginners Guide to Uncertainty in Measurement, Stephanie Bell, Measurement Good Practice Guide No. 11, Issue 2, 2001, NPL, UK
  2. M3003. The expression of uncertainty and confidence in measurement. UKAS, Feltham, Middlesex, TW14 4UN, UK
  3. Guide to the Expression of Uncertainty in Measurement (GUM), 1993, International Organization for Standardization (ISO), Geneva

Reporting of results

Any calibration is only valid on the date of calibration and for the environmental conditions which existed during the calibration. When the result of the calibration of a hydrophone is reported, the environmental conditions which pertain to that calibration must be stated. These include all those conditions which may influence the sensitivity of the device. It is recommended that the conditions reported should include:

  • date of calibration
  • water temperature
  • depth of immersion (or applied hydrostatic pressure)
  • type of mount or rigging used
  • length of soaking time and any wetting procedure adopted
  • orientation of the transducer about any axis or alignment mark and whether the alignment was done manually or acoustically
  • maximum acoustic pressure experienced by the hydrophone
  • any assumptions made about the device under test (eg the position of the acoustic centre)

If use is made of a calibrated hydrophone in an environment significantly different to that which existed during calibration, the user may need to increase the calibration uncertainties to account for the change in environment.

Recommended recalibration periods for hydrophones

For reference hydrophones that are used purely for calibration purposes, it is recommended that a recalibration be performed annually. Where hydrophones are used in the field and potentially may be subjected to abuse, calibrations may be required at shorter intervals.

More detailed information on uncertainties can be found on page 2.

For further information, contact Stephen Robinson

Last Updated: 19 Oct 2015
Created: 6 Jun 2007


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