Array response & antenna theory
Learning objectives
- Define the array factor |H(θ)| for a uniformly-spaced linear array of N elements
- Explain the main lobe, side lobes, and grating lobes in antenna terms
- Apply steering: shift the main lobe to direction θ₀ by progressive delays
- Choose element spacing d < λ to avoid grating lobes
Put N sources (or N receivers) in a line, space them by d, and add their outputs. You have a beamformer: a compound sensor that is more sensitive to some directions than others. The physics is identical to antenna design, which is why seasoned acquisition engineers are secretly antenna engineers.
The array factor
For a plane wave arriving at angle θ from broadside, the normalized array factor of N uniform elements spaced d apart is:
A main lobe sits at the arrival direction (θ=0 for broadside); side lobes sit at regular angles away from it, with ~13 dB suppression for uniformly-weighted arrays; and if d grows past λ, grating lobes appear — additional main-lobe-strength peaks at other angles, which ruin the array’s discrimination.
Steering, beamwidth, and the d < λ rule
By adding progressive delays Δt = (d sinθ₀)/V to successive elements you can shift the main lobe to any direction θ₀. Beamwidth is roughly λ/(N d) radians for the main lobe, so more elements or larger total aperture give a narrower beam. The rule d < λ / (1 + |sinθ₀|) prevents grating lobes.
Every source-array design (air-gun array, vibroseis sub-arrays) and every receiver-array design (hardwired geophone groups, DAS gauge-length averaging) is ultimately a beamformer. Get the array factor right in your head first; the rest is detail.
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.