Douze and Laster (1979) considered the problem of identifying the mode of propagating isotropic seismic noise by the characteristics of the cross-spectrum between two geophones. For the vertical component of isotropic single-mode Rayleigh wave noise, they derived the theoretical relation
where Ψ(ω) is the power spectrum, Ψ12(ω) is the cross-spectrum between geophones 1 and 2, Δx is the geophone separation, V is the phase velocity of the waves at angular frequency ω, and J0 is the zero-order Bessel function. They noted that the cross-spectrum is purely real and “no apparent phase velocities can be obtained from the data.”
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