The effect of geophone arrays on random noise

Abstract

For spatially random noise consisting of plane surface waves arriving at a geophone from all directions with equal probability, I compute the noise‐to‐signal ratio (N/S) as a function of the ratio of geophone spacing Δ to noise wavelength λ for linear arrays of 12, 24, 36, and 72 vertical and horizontal geophones. For arrays of M vertical geophones, deviation of N/S from that for pure random noise, [Formula: see text] is marked. As Δ/λ increases, N/S decreases, attaining [Formula: see text] for a value of Δ/λ that defines a coherence distance. As Δ/λ increases beyond the coherence distance, N/S continues to decrease slightly, only to increase again as Δ/λ approaches unity. For linear arrays of 12 to 72 vertical geophones, Δ/λ at the coherence distance is nearly independent of M and is about 0.32. The value of Δ/λ for which N/S = 0.5 changes with M but L/λ, L being the array length defined as MΔ, is independent of M. Introducing a random 14 percent variation into the sensitivities of the individual geophones has a small influence on the coherence distance and the N/S = 0.5 point when the N/S curves for 50 arrays are summed. Introducing a random 14 percent variation into the spatial coordinates of the geophones changes the coherence distance and 0.5 point only slightly but reduces the amount by which N/S fails below [Formula: see text] For arrays of horizontal geophones, there are two N/S curves depending upon whether the horizontal motion is in the plane of incidence of the signal (SV waves) or perpendicular to the plane of incidence (SH waves). Due to the superdirectivity of the array, N/S for SV waves incident on an SV‐wave geophone array is very small for values of Δ/λ greater than about 0.1. The N/S curves for SH waves incident on an SV‐wave array resemble those for the arrays of vertical geophones. Again there is a coherence distance, but the Δ/λ value is twice that for the vertical geophones, indicating the geophones should be separated by about two‐thirds of a wavelength.