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Crosscorrelation structures in the Green function retrieval bycrosscorrelating wavefields are revealed using rigorous mathematical theory onintegral equations. The previous practice on extracting the Green function bycrosscorrelating the wavefields recorded at two locations produced by the samesource and then summing such crosscorrelations over all sources on a boundaryis inadequate and has limitations in recovering the low frequency content inthe Green function. To recover the exact Green function, we need crosscorrelatethe wavefields observed at the two locations generated by different sources,respectively. The crosscorrelation structure can be viewed as a matrix in thediscrete case or an integral operator in the continuous case. The previousGreen function retrieval method corresponds to the identity matrix multiplyinga constant. If the matrix is diagonal, the wavefield crosscorrelation is stillwithin the wavefield due to the same source but different weighting should beapplied for different sources. We have derived analytically thecrosscorrelation kernels for two important cases, the plane boundary and thecircular boundary, and both kernels are symmetrical difference kernels, whichcorrespond to convolutional operations. For high frequency waves or far-fieldsources, the kernels reduce to the know result which is the Dirac deltadistribution (or an identity matrix). For other boundaries of general geometricshapes, numerical schemes like the boundary element method can be used to solvefor the kernel matrix.

Crosscorrelation kernel in the Green function retrieval and time reversal acoustics