Air-gun arrays: bubbles, ghosts, tuning

A single air gun fires and its compressed air escapes into the water as a bubble. The bubble overshoots, expands until the water stops it, collapses and expands again, and every collapse sends out a pressure pulse. So behind the sharp primary comes a train of bubble pulses lasting a few hundred milliseconds, the bubble train. Left alone, it rides behind every reflection like a slow echo.

The Rayleigh-Willis formula

The bubble period of a single gun follows the Rayleigh-Willis formula. With firing pressure $P$ in psi, chamber volume $V$ in cubic inches and tow depth $h$ in metres,

where $K$ is an empirical constant for the type of gun and the 10 is the depth of water that weighs one atmosphere, so $h + 10$ stands for the hydrostatic pressure. The figure below uses $K = 0.0150$: at 2000 psi and 6 m a 150 in³ gun has $T\_b \\approx 100$ ms and a 40 in³ gun about 64 ms. The period grows only as the cube root of the volume, so four times the air lengthens it by about 60 %, and towing deeper shortens it.

The figure opens on six identical 100 in³ guns. Spread their sizes with the volume range, watch the bubbles in (b) fall out of step while the primaries stay together, and read the primary-to-bubble ratio in the table; then change the number of guns and the tow depth.

Array tuning

Identical guns share one bubble period, so their bubbles add as coherently as their primaries: six of them give the same primary-to-bubble ratio as one, 1.8 in the figure. Spread their volumes over a range of four and the periods fan out from 67 to 106 ms. The primaries still add at the firing instant, the bubbles no longer do, and the ratio rises to 13.1 for a loss of only 3 % in peak-to-peak strength. A spread needs guns to fill it: three guns over a tenfold range leave each bubble on its own and reach only 4.6, where twelve over the same range pass 20. That is why a production source array carries 18 to 48 guns in three to six strings, sized from about 20 to about 250 in³, often clustered in pairs whose bubbles interfere as well (the figure leaves clustering out).

Measuring the result

Array specifications describe the far-field signature straight below the array, band-limited to 128 Hz, by two numbers (Dragoset, 1990). The peak-to-peak strength, in bar·m, runs from the primary peak to the trough of its ghost. The primary-to-bubble ratio divides the primary’s peak-to-peak by the largest peak-to-peak in the bubble train, measured, like the primary’s, over a window just long enough for one pulse and its ghost. A single gun manages about 2 and a tuned array several times more. Because both numbers come from the far field, they include the ghost: switch the sea surface off in the figure and the tuned array’s ratio falls from 13.1 to 6.7.

What depth does to the spectrum

Like the dynamite charge of Section 1.2, the gun sits below a free surface that reflects with coefficient $-1$, so the far field carries a ghost $2h/c$ behind the primary and, straight down, the spectrum is multiplied by $2\\,|\\sin(2\\pi f h/c)|$. The first notch above zero falls at $f\_g = c/(2h)$, or $750/h$ Hz for $c = 1500$ m/s: 125 Hz at 6 m, above most of the useful band, and 50 Hz at 15 m, inside it. The same factor sets the low end: at 5 Hz the ghost costs 12 dB at 6 m but only 4 dB at 15 m. Deep tow buys the low frequencies that full-waveform inversion wants at the price of the high ones. It also lowers the tuned array’s ratio, from 13.1 at 6 m to 5.0 at 15 m, and mostly through the ghost rather than the bubble periods. At 6 m the ghost follows each bubble pulse by only 8 ms, overlaps it and trims it; at 15 m it comes 20 ms late, and every bubble and its inverted ghost stand apart at full height. Holding the bubble periods at their 6 m values, the later ghost alone takes the ratio to 7.0, while the shorter periods alone would only take it to 12.7.

References

• Dragoset, W. H. (1990). Air-gun array specs: a tutorial. The Leading Edge, 9(1), 24-32.
• Ziolkowski, A. (1970). A method for calculating the output pressure waveform from an air gun. Geophysical Journal of the Royal Astronomical Society, 21(2), 137-161.
• Landrø, M., and Amundsen, L. (2010). Marine seismic sources part I. GEO ExPro, 7(1).
• Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.