Array response & antenna theory

Put $N$ sources or $N$ receivers in a line, $d$ apart, and add their outputs. The result is a beamformer: one compound sensor that responds more strongly to waves from some directions than from others. The mathematics is the same as for a radio antenna, which is why array design in acquisition borrows its words from antenna theory.

The array response

A plane wave arriving at angle $\\theta$ from broadside (the normal to the line) reaches element $n$, at $x\_n = nd$, a time $x\_n\\sin\\theta/V$ earlier than the element at $x = 0$ (the far end hears it first when $\\theta > 0$). Delay each output by $x\_n\\sin\\theta\_0/V$ and add them with weights $w\_n$ that sum to 1, and a wave from direction $\\theta$ comes out with amplitude

the Fourier transform of the weights. For equal weights it is $|\\sin(N\\pi u)/(N\\sin\\pi u)|$. The main lobe sits at the steered direction $\\theta\_0$. Side lobes sit on either side of it; with equal weights the strongest is 11.3 dB down for four elements and approaches 13.3 dB down as $N$ grows. And because $|H|$ repeats every time $u$ grows by 1, a wide spacing lets a full-strength copy of the main lobe, a grating lobe, point in another real direction.

The figure opens with eight elements 0.90 wavelengths apart, steered to 20°, and (b) shows the problem: a grating lobe at −50.3° as strong as the main lobe. Close the spacing until it leaves, then steer, add elements and taper the weights to see what each one buys and what it costs.

Steering, beamwidth and the grating-lobe rule

Plate (c) puts the whole story in one picture. The response repeats every 1 in $u$, but only directions with $|\\sin\\theta| \\le 1$ exist, so the real directions form a window of width $2d/\\lambda$ centred on $u = -(d/\\lambda)\\sin\\theta\_0$. Widening the spacing widens the window, and steering slides it. A grating lobe appears when the window reaches the repeat at $u = \\pm 1$, so its peak stays out while $d < \\lambda/(1 + |\\sin\\theta\_0|)$: 0.745 wavelengths at 20°, one wavelength at broadside, and half a wavelength for steering all the way to the end of the line. The flank of the lobe arrives first, which is why exercise 1 finds the last of it near 0.65, $(1 - 1/N)$ times that limit.

Steering needs no hardware, only a delay of $\\Delta t = d\\sin\\theta\_0/V$ per element (12.3 ms for the figure’s 72 m spacing at 2000 m/s and 20°), but it widens the main lobe by about $1/\\cos\\theta\_0$. At broadside the half-power beamwidth is about $0.886\\,\\lambda/(Nd)$ radians: the aperture in wavelengths sets it, so at half a wavelength four elements give 26.3° and sixteen give 6.4°. A taper trades the other way: the Hann weights of exercise 4 push the strongest side lobe from −13.1 to −31.5 dB and pay with a main lobe 1.5 times wider.

Every source array (air-gun clusters, vibroseis groups) and every receiver array (hard-wired geophone groups, and in continuous form the gauge length of DAS) is such a beamformer. A grating lobe is spatial aliasing by another name: the repeat every $1/d$ in wavenumber that section 0.6 meets in a line of receivers.

References

• Vermeer, G. J. O. (2002). 3-D Seismic Survey Design. SEG Geophysical References 12.
• Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.
• Cordsen, A., Galbraith, M., Peirce, J. (2000). Planning Land 3-D Seismic Surveys. SEG Geophysical Developments 9.
• Vermeer, G. J. O. (1990). Seismic Wavefield Sampling. SEG Geophysical Monograph 4.