Source-array directivity & ghost notch

Part 1 — Sources

Learning objectives

Combine array factor (§0.5) with ghost factor (§1.2) into a single 2D radiation pattern
Choose element spacing d so directivity lobes don't fall on your target bearings
Select submergence depth h so the ghost notch falls above your usable band
Recognise the f-θ response surface as an inescapable design constraint

Part 0.5 taught the array factor. Part 1.2 taught the ghost factor. A real source array has both — and the total radiation pattern is their product.

|H(\theta, f)| = \underbrace{\left|\frac{\sin(N \pi d \sin\theta / \lambda)}{N \sin(\pi d \sin\theta / \lambda)}\right|}_{\text{array factor}} \;\times\; \underbrace{\bigl| 2\sin(2\pi f h \cos\theta / V) \bigr|}_{\text{ghost factor}}

What the interaction looks like

At vertical incidence (θ = 0), the array factor is 1 and the ghost factor gives the familiar comb filter: notches at f = n·V/(2h). At grazing angles (θ → 90°), the ghost factor’s cosθ term moves the notch to infinity and only the array factor matters. So the array "sees" different notch frequencies in different directions — a fact that complicates near-offset recording of shallow events.

Design rules of thumb

Spacing d < λ_min avoids grating lobes that would leak energy out the sides.
N d ≈ 1-2 λ gives a well-defined vertical lobe without being over-narrow.
Submergence h places the first ghost notch below your lowest usable frequency when possible, or above your highest when not.
Asymmetric arrays can be designed to steer or broaden the main lobe; rarely used, but available.

References

Dragoset, B. (1990). Air-gun array specs: a tutorial. Geophysics, 55(11), 1426–1440.
Ziolkowski, A. (1970). A method for calculating the output pressure waveform from an air gun. Geophysical Journal International, 21(2), 137–161.
Vermeer, G. J. O. (2002). 3-D Seismic Survey Design. SEG Geophysical References 12.
Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.