As anybody will notice when becoming involved in seismic source modelling for the first time, the physical units can be a little confusing.
The problem is that the modern SI unit system is mixed somewhat arbitrarily with old imperial measures, and it is common for example to measure distance in metres but pressure in pounds per square inch and volume in cubic inches.
For reference, note the following:-
| 1 Newton | Unit of force, 1kg. m. s-2 |
| 1 Pascal | Unit of pressure, 1 Newton m.-2 |
| 1 bar | Unit of pressure, 105 Pascal |
So the traditional pressure for airguns, 2000 pounds per square inch converts to bars as follows (2.2 pounds = 1kg; 3.28 feet = 1m; 12 inches = 1 foot; g = 9.8 m.s-2):-
psi = (pound)/(in2) = (12 3.28)2 / 2.2 kg. m-2 = 9.8 (12 * 3.28)2 / 2.2 Newtons m-2 = 6901.02 Pascal
So 2000 psi = 6901.02 2000 Pascal = 6901.02 2000 / 100000 bar = 138 bar approximately.
Exploration geophysicists conventionally compare airgun signatures in terms of simple time-domain characteristics such as peak to peak pressure and primary to bubble ratio. The peak to peak is used for signatures in the far-field but referred back to 1m from the source so it is quoted in units of bar-m. If the signature is measured at some hydrophone position, the units used are bars. To refer these back to 1m, simply divide each pressure value by the distance in m. between the hydrophone and the source. This peak to peak output is critically dependent on the bandwidth in which the signature is measured and it makes little sense to quote a peak to peak figure without this bandwidth. As an example, a reasonably powerful airgun array will have a peak to peak of perhaps 50 bar-m in a bandwidth of 0-128Hz. If this were measured in 0-256Hz, it might be more than 50% larger.
This effect does not continue indefinitely however. Airgun signatures have very little energy above 10kHz so that measuring the peak to peak in a 10kHz bandwidth and a 20kHz bandwidth, very little difference will be seen.
Marine biologists differ in two ways in how they assess pressure. First, they generally deal with much quieter sounds (although a Beluga whale's call is astonishingly loud) and commonly use the microPascal (muPa) = 10-6 Pascal as a measure of pressure. Second, they are much more interested in the spectral distribution of the acoustic field and since mammal hearing behaves in a complicated logarithmic manner, measures of acoustic output are often given in terms of dB (decibels) relative to some base value. For example, marine environmental impact reports often use units such as dB relative to 1muPa / Hz at 1m, dB relative to 1muPa at 1m or dB relative to 1muPa 2-s at 1m. The first two of these are related to the amplitude spectrum and the third to the energy spectrum.
This is elegantly described by Fricke, Davis and Reed (1985) “A standard quantitative calibration procedure for marine seismic sources”, Geophysics, 50(10), p. 1528-1532. They proposed calibrating the amplitude spectrum by referencing it to 1 muPa / Hz at 1m. This naturally makes spectral comparison independent of sample interval, time-series window length and hydrophone measurement position providing a consistent way of comparing signatures.
This can be calculated from muPa/Hz simply by integrating over some window in the frequency domain, perhaps a 1/3 octave or full octave window. More typically, it is a measure of a time domain property such as zero to peak or peak to peak, measured in muPa and converted to dB.
This is calculated by summing the squared pressure values (either referenced to 1m or at some distance from the source) over one second and converting into dB. For a transient like an airgun signature, it is normally calculated over a window in which the total energy rises from 5% of its total value to 95% and is then normalised to 1s. duration. It is very closely related to the energy density output in Joules m -2 and can be calculated from it by multiplying by (ρc/4π) where ρ is the density of sea water and c is the velocity of sound). If it is quoted /Hz instead of -s, it is measured from the mean squared pressure per unit frequency.
1 cu.in. = 2.54 2.54 2.54 cu cm. = 0.01638704 litres. (1000 cu.cn. = 1 litre).
So a 300 cu.in. airgun has a volume of around 4.92 litres.
The SI unit for energy is the Joule, which is 1 Newton m., (note that the SI unit for power is the Watt which is 1 Joule s-1). For example, the acoustic energy given out by a typical 7 airgun sub-array of around 700 cu.in would be around 30,000 Joules depending on the overall energy interaction in the array. This represents around 5-40% of the total energy depending on the interaction in the array. The rest of the energy is dissipated as heat.
Since Gundalf 6.1c, the amplitude spectrum in the full report can be annotated not only with the pressure, (measured in dB. relative to 1muPa/Hz. at 1m. As described above), but also optionally with energy flux density, following the SEG standard as reported by Johnston, R. C., Reed, D. H. and Desler, J. F., 1988, “Special report on marine seismic energy source standards”, Geophysics, 53, no. 4, 566-575. Energy flux density is effectively p^2/(ρ*c), where p is the pressure in Pascals, ρ is the density of the medium, in most cases sea water, and c is the velocity of sound in that medium. Here, ρ is taken as 1025 kg./m^3 and c as 1500 m/s in this calculation and the resulting unit for energy flux density is dB. relative to 1 Joule/m^2/Hz. at 1m.
Gundalf uses the strange mixed world of distance in metres, pressures in psi and volumes in cu.in to align itself with common practice although internally it uses SI units, (formerly known as MKS - Metre, Kilogram, Second). In normal reports, Gundalf quotes peak to peak and primary to bubble ratio amongst other time-domain characteristics, but for amplitude spectra and marine environmental reports, it uses units which are familiar to marine biologists as described above.
To compare time-domain sources, they are referred back to the pressure as it would have been at 1m. from the source, (assuming that it is a point source) as described above. The resultant unit is the bar-m. which is therefore a distance independent measure of pressure for a seismic source, (since the amplitude of a spherical source decays as 1/r, multiplying the pressure at a distance r from the source by the value of r gives a distance independent value.
It is also possible in Gundalf to specify a measurement position. In this case, the resulting pressure field is shown in bars.
As described above, the standard work on this is by Fricke, Davis and Reed (1985) “A standard quantitative calibration procedure for marine seismic sources”, Geophysics, 50(10), p. 1528-1532. They proposed calibrating the amplitude spectrum by referencing it to 1 muPa / Hz at 1m making spectral comparison independent of sample interval, time-series window length and hydrophone measurement position providing a consistent way of comparing signatures. Gundalf uses this option exclusively in generated reports.
Prior to 2018, Gundalf followed what appeared to be industry practice and a 1988 SEG standard (Johnston et. al.) in only using positive frequencies when calculating amplitude spectra. Unfortunately, this is ambiguous in both Fricke et. al. (1986) quoted above and also Johnston et. al. (1988) and the worked examples they give are not sufficient to resolve this. Unfortunately, amplitude spectra calculated like this do not satisfy Parseval's theorem, a fundamental theorem of Fourier analysis and give values 3dB lower than Parseval- compliant scaling. (Parseval's theorem states that the sum of the squared time values (measured in bar-m) scaled by the temporal sample interval should be the same as the sum of the squared amplitude spectral values (measured in dB re 1microPa/Hz at 1m) scaled by the frequency sampling interval.) Whether the SEG standard actually requires this or not is therefore a moot point so in 2018 the decision was taken to adhere to Parseval's theorem and the amplitude spectra scaled accordingly. All Gundalf versions since have followed this principle as environmental modelling has grown in importance.
However, there may be some existing practice still using historic so-called SEG standard amplitude spectral scaling. To allow for this, versions of Gundalf from the Cloud version 8.3j (Nov 2023) onwards provide the option to do either, although the default remains Parseval compliant. Output reports and amplitude spectral headers are annotated to indicate which option is in place using the phrases "SEG amplitude spectral scaling" or "Parseval amplitude spectral scaling", although we repeat that it is a moot point as to whether this was actually the case in the SEG standard. It is certainly true that some historic practice (including old versions of Gundalf) did this, but this change makes it explicit and your choice.
If you can, quote your source strength in bar-m and your primary to bubble in a specified bandwidth. Make sure you state whether this is is zero to peak (for example for near-fields) or peak to peak (far-fields). Without the bandwidth, the numbers don't mean very much. Alternatively, you can quote the source strength in Fricke units of dB relative to 1 muPa/Hz at 1m. This measure is independent of bandwidth or duration.
Marine biologists nowadays use SPL (Sound Pressure Level) and SEL (Sound Energy Level) almost exclusively. SPL is essentially the zero to peak or peak to peak quoted in dB. relative to 1muPa at 1m, rather than in bar-m as geophysicists most commonly used. Again the assumption is that the bandwidth of measurement is wide enough to give a stable measurement.
SEL is a little more difficult to understand. It is a truncated form of the rms pressure with units of dB. relative to 1 muPa2-s at 1m. but is very useful for estimating cumulative exposure of animals to sound and the resultant effects on TTS (Temporary Threshhold Shift) and PTS (Permanent Threshold Shift).
It is worth noting finally that both SPL and SEL are essentially time-domain measures with a non-trivial relationship with the corresponding signature spectra.