Dynamite: charge, depth, coupling

Part 1, Sources

Learning objectives

Explain why the depth of the charge, and the slow weathering above it, set the ghost notches, while the charge size does not
Calculate the ghost delay $\tau$ and the notches $f_{\mathrm{notch}} = n/\tau$ from the layers above the charge, and recognise $V/(2h)$ as the uniform-ground case
Describe the depth trade-off: below the base of the weathering for coupling, not so deep that the comb of notches crowds the band
Predict what a larger charge does to the wavelet: amplitude up as $W^{1/3}$ , peak frequency down as $W^{-1/3}$

A dynamite charge fired at depth $h$ sends one spherical wave out. Its down-going half is the primary, the wave we want. Its up-going half reaches the free surface, where the stress must vanish, and comes back down inverted: the ghost. The ghost follows the primary late by twice the one-way vertical time from the charge to the surface,

\\tau = 2\\int\_0^h \\frac{dz}{V(z)},

and in frequency it is a comb filter, $|G(f)| = 2|\\sin(\\pi f \\tau)|$ , with a notch at every multiple of $1/\\tau$ :

f\_{\\mathrm{notch}} = \\frac{n}{\\tau}, \\quad n = 1, 2, 3, \\dots

In uniform ground $\\tau = 2h/V$ and the first notch is $V/(2h)$ : a shot at 12 m in rock of 600 m/s notches at 25 Hz, at the bottom edge of the band a 1 kg charge puts out. Real holes cross a slow weathering layer, a few metres of loose soil and rock at 400 to 800 m/s, into faster bedrock, and the weathering dominates the delay. In the figure, move the charge down its hole and through the base of the weathering, then change its size, and see which of the two moves the notches in (c).

The figure opens on a typical column, 8 m of weathering at 600 m/s over bedrock at 2000 m/s, with a 1 kg charge at 12 m. The ghost is 30.7 ms late, so the first notch falls at 32.6 Hz and the second at 65.2 Hz, both inside the 24.1 to 81.8 Hz band the charge puts out. Every metre of weathering above the charge adds 3.3 ms to the delay, every metre of bedrock only 1.0 ms. Drilling deeper never clears the band: at 30 m the first notch drops to 20.5 Hz, below the band, but its multiples still fall every 20.5 Hz and two of them cut through it; in uniform 600 m/s ground at 30 m they fall every 10 Hz, straight through the band. A notch-free band needs a short delay, which means little slow rock above the charge. A deep ghost does repay part of the cost at the low end: in the typical column at 30 m it lifts 10 Hz by 6 dB, where a shallow ghost cuts it, though in uniform ground at 30 m a notch falls exactly on 10 Hz.

What the charge size controls

The charge sets the wavelet, not the ghost. Explosions obey cube-root similarity scaling: the pulse from a charge of mass $W$ is the pulse of a reference charge stretched in time by $W^{1/3}$ . Its peak frequency therefore falls as $W^{-1/3}$ , and its far-field peak amplitude grows only as $W^{1/3}$ . Doubling the charge adds about 2 dB to the peak pressure and lowers the peak frequency by a factor of 0.79; eight times the charge doubles the amplitude (+6 dB, the peak amplitude in the figure’s table) and halves the frequency, from 50 to 25 Hz in the figure. The spectrum grows faster than the peak, because the stretched pulse also lasts longer: near its peak it rises as $W^{2/3}$ , about 4 dB per doubling, so plate (c) shows the 8 kg charge 12 dB above the 1 kg one. The low frequencies gain the most. The notches do not move at all, because they depend only on the column above the charge.

Coupling

Coupling is how efficiently the explosion puts its energy into elastic waves rather than into crushing loose soil or venting gas up the hole. A charge in the dry, unconsolidated weathering couples poorly and feeds ground roll; a charge in competent, preferably water-saturated rock below the base of the weathering couples well. The hole must also be tamped: an untamped hole blows out, a hole that is too shallow craters, and both waste energy. The field practice follows from the figure: drill through the weathering and a few metres into the rock below it, which buys the coupling, and stop there, because every further metre lowers the ghost notch and gains little.

References

Sharpe, J. A. (1942). The production of elastic waves by explosion pressures. I. Theory and empirical field observations. Geophysics, 7(2), 144-154.
Ziolkowski, A., Lerwill, W. E. (1979). A simple approach to high resolution seismic profiling for coal. Geophysical Prospecting, 27(2), 360-393.
Ziolkowski, A. (1993). Determination of the signature of a dynamite source using source scaling, Part I: Theory. Geophysics, 58(8), 1174-1182.
Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.
Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Cordsen, A., Galbraith, M., Peirce, J. (2000). Planning Land 3-D Seismic Surveys. SEG Geophysical Developments 9.